MATHEMATICAL SCIENCES VIGRE SEMINAR
MATHEMATICAL SCIENCES VIGRE SEMINAR CAAM 699
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Date Created: 10/19/15
Model Reduction of Electrical Circuits Timo Reis Technische Universit t Berlin Model Reduction Seminar CAAM Rice University 091 82008 1972 800 KHZ Intel 8008 10 3500 comp 1982 Intel 286 15M 134000 comp 12 MHz 2 km decrease of feature size increase of chip complexity increase of operation frequencies increase of interconnect length modelling thermal electromagnetic effects gt 73105 2002 Intel Pentium 4 013 55 1o6 comp 24 GHz 4 km 109 Circuit simulation in industry automatic generatio of circuit model Vnetwork list circuit simulation package eg SPICE TITAN simulation results Circuit simulation in industry within model reduction automatic generatio of circuit model Circu network list circuit model reduction circuit simulation acka e j eg SPICE TITAN approximating small network list simulation results Task uv2 t linear electrical circuit with resistances capacitances inductances ideal transformers Task approximating linear electrical circuit with fewer resistances capacitances inductances ideal transformers Motivation equations Modell edu on Example Suml39 Overview a Circuit equations and descriptor systems 9 Model reduction 9 Circuit synthesis 9 Example 6 Summary and Outlook 6 References circuilequalions Modelredu on xample Summary lOutloer Circuit model Given 0 voltages at voltage sources uV Rt a RW 0 current at currents sources i it a R Wanted 0 voltages and currents at resistances URU pa 0 voltages and currents at capacitances uot 01 O voltages and currents at inductances uLt M1 0 voltages and currents at ports of ideal transformers unU Unt 71 I39nU currents at voltage sources ut 00 voltages at current sources Mt xample Summary iCiutloer circuit equations Model redu on Circuit model Component relations 0 URU FiI39RU resistances 0 01 C uoa capacitances 0 uLt L 10 inductances Una T 1170 iTf T7 I39nf ideal transformers with RI resistance matrix symmetric positive definite C capacitance matrix symmetric positive definite L inductance matrix symmetric positive definite T transformer matrix Motivation circuitequations Modelreduction Gil Example Suminaryanduutlonk References Circuit model cont Kirchhoff s current law ARIR1 Aoic1 AULU ATiiTiU ArriTrU AVAt Arit 07 where AR A0 A An An AV Ai element incidence matrices eg AR awwith 1 j th resistance begins in node i 3 71 j th resistance ends in node i O else Motivation circuilequalions a Example Sumlnaryand mlonk References Circuit model cont Kirchhoff s voltage law um A em uom A em um Aha um Aha um AW una AW um ACeU Motivation circuilequalions Modelreduction Circuiisynlhesis Example Summaryand ullonk Relerences Overall circuit model ACOAC o o o em iAFeFl lA iAL ATATT iAv em 4 g o L o 0 Mr A o o o W o o w 0 0 0 0 mm A7TTA 0 0 0 mm o e we 0 0 0 0 Mr A 0 0 0 WU 0 fl Motivation circuilequalions Model reduction Example Sumlnaryand Uullonk Relerences Overall circuit model ACCAZ o o o em iAFeR M iAL ATATT iAv em 4 g o L o o w L o o o w o o w 0 0 0 0 m0 47774 0 0 0 mm 0 0 mo 0 0 0 0 Mr A 0 0 0 W0 0 7 em um 7 7A o o o W W 7 o o 7 mt Motivation circuilequalions Modelreduction Clruuhsyn ens Example Summaryand ullonk Relerences Overall circuit model ACME 0 0 039 em I 39 RR V AL 4riAnT Av em 39I 7A o 39l o L o o W A o W o o w l 0 0 0 nJlmmJ 4474 0 0 0 mmH o 0 WW 0 0 0 0 Mr A 0 0 0 W 9 um 42 3 Motivation circuilequalions Modelreduclion Cll Example Summaryand ullonk Relerences Overall circuit model descriptor system circuit equations Problem of circuit model reduction Incidence and component matrices AR Ac AL Am An Av Ah R 07 L T of some circuit 8 Incidence and component matrices ZR 2 0 A I m I m Z v Z h R C I 7 of some smaller circuit g such that the currentvoltagebehaviour at the sources of S and S are approximately the same circuit equations Reformulation Circuit descriptor system EXt Axt But W CXU with EA 6 11W 3 07 6 RM Wanted Find Z lt n and a circuit descriptor system E742 2W Eua 71 5m with Eli e R E N27 6 R such that for all sufficiently smooth u e L2R holds 7 7y is small Motivation circuilequalions Modelreduc on 7 Example Summaryandamlonk References Descriptor systems EXt Axt But W CXU Motivation circuilequalions d 39 Example Summaryandamlonk References Descriptor systems WETTquotX1 WATTquotX1 WBut W CTT 1X1 circuit equations Model reduciion Cu cm Motivation Descriptor systems 5 mam ii Kronecker normal form 0 N nilpotent with N O Wquot 7 O u quotKronecker index Is Example Sumlnal yand uilook Reierences circuit equations Model reduction Cu cm Motivation Descriptor systems 5 iiii2Eiiii ii Kronecker normal form 0 N nilpotent with N O Wquot 7 O u quotKroneoker index 0 solution with homogeneous initial conditions p71 yt 01 8r TmB1u7 dr 7 c2 ZNKBZWU 9 KO Is Example Summaryandautlook Reierences circuit equations Model reduction Cu cult Motivation Descriptor systems 5 iiii2Eiiii ii Kronecker normal form 0 N nilpotent with N O Wquot 7 O u quotKroneoker index 0 solution with homogeneous initial conditions p71 yt 01 8r TmB1u7 dr 7 c2 ZNKBZWU 9 KO 7 I O 1 7 1 I O H H 0737W0 OJW undP7T O OJT spectralprOJectors Is Example Summaryanduutlook Reierences Motivation circuilequalions d 39 Example Summaryandamlonk References Motivation circuilequalions Modelreduction Cucth Is Example Sumlnaryand urlonk Relerences Properties of circuit descriptor systems 91 iii MM yltoiramp ii Mt Motivation circuilequalions Modelreduction Cucth Is Example Sumlnaryand urlonk Relerences Properties of circuit descriptor systems 91 39 f 39 f 39 f X 150 u v3 WhigE3 I39VI Motivation Model reduction Cu cur Is circuit equations Example Summary and Outlook References Properties of circuit descriptor systems 91 39 f 39 f 39 f X 150 u v3 WhigE3 I39VI r r 1 W 1LLt rr0r u TR uRTdT 0 energy lost by the system circuit equations System is passive ie for all sufficiently smooth u e 00R t e R holds r umTym ch 2 o 700 Motivation circuilequalions Modelreduclion 7 Example Sumlnaryandaurlonk References Conclusion Model reduction method should preserve passivity Results in circuit realization theory Any passive system can be realized as some electrical circuit 0 Cauer1920s1940s Brune1930s1940s o Darlington 1950s1970s 0 Anderson Vongpanitlerd 1960s1990s Moiivaxion Inequations Modelreduction rcunsynu lesis Exampm Suurnnaryan Iook References Contents 9 Model reduction Model reduction Problem formulation Given Passive descriptor system EXI Axt But with B 07 e 11W EA 6 HM Wanted Passive descriptor system N39 Ilia Eua with E N27 6 1R Ell e R Z lt n such that For all possible input u the output difference y 7 N should be small Model reduction Energybased ansatz Minimal energy that has to be put into the system to steer to the state Xi U Egan inf 7 uT7yTdT ue 01RmLzR 207 andiijosteers to X0 X0 Maximal energy that can be extracted from the system initialized by x EabX0sup Amun ynwm uec RmLzR 20 and initial condition X0 X9 Idea of model reduction Truncate those states X9 6 R for which E5XQ is big and at the same time EabXn is small Motivation equations Modelreduction Circu Example Sumlnaryand utlook References Energybased ansatz for descriptor systems R Stykel The energy functionals have the representation Em XJXWQ Ewe XJETyEXn where X and 31 are solutions of projected Lur e equations MET E20 2525 7322 7DBBT7JT 72160160773 widen 43719 1cm 5ch e pBMUT emchJ ETJ B WOTViv PrT CvTJw lngVIQT JCJCT extlimping X 73203 2 o y Wyn 2 o where 0 M0 2 lim CsE7A Boris W o It guise O 73 and 73 are spectral projectors of E 7 2t Motivation Ciicuiiequations Modelreduction Cucth Is Example Summaryand uilonk Reierences Model reduction methods R Stykel Balancing Find invertible W T e R such that for r rankE holds I o 7 2i 0 WET O O WAiBCT7O 2 00 W37 Bf CTC c i 7 8007 i f gt0 W D KW 1 T yTTdiag OHM Xdiag 71a With 71 Z Z a Summary and Umlonk Relerences Motivation Cucuhequations Modelreduction Cucth Is Example Model reduction methods R Stykel balancing und partitioning K 0 0 E11 E12 0 WET 0 WK 0 WAiBCT A21 A22 0 o o o o o RX 31 WB 32 CTC1 02 cm Boo x1 0 o W TXWquot o 2 o o o o Summary and Umlonk Relerences Motivation Cucuhequations Modelreduction Cucth Is Example Model reduction methods R Stykel balancing und partitioning K 0 0 E11 E12 0 WET 0 WK 0 WAiBCT A21 A22 0 o o o o o RX 31 WB 32 CTC1 02 cm Boo x1 0 o W TXWquot o 2 o o o o Motivation equations Model reduction Circu Example Summary and Umlook References Model reduction methods R Stykel Balancing und truncation Motivation Circuiiequations Modelreduction Cucth I Example Summary and Umlonk Relerences Model reduction methods R Stykel Balancing und truncation o o Ax B BB Cc1 cw 7 x1 0 tie oi Further properties of the model reduction method Theorem Stykel R If the inputoutput mapping is bounded then for all u 6 L201 holds HY39 y39HL2RJ S CHMMeme with n Jlt 2 1 2 CHM0MUTH2 Z lt1220kgt HVJlt1 Motivation 39 Modelreduction Cll Example Surnlnaryanduutlonk Relerences Numerical aspects 0 solution of Lur e equations with Newton s Method R Sachs Stykel in progress 0 singularvaluebased balancing algorithm Stykel 0 Lur e equations via eigenspaces of ext Hamiltonian matrix pencils in progress Special properties of circuit equations O explicit formulas for 73 73 and Mg 0 solutions of the Lur e equations fulfill X 8318 where S diagl fl 0 further exploitation of structure of circuit equations is possible 0 circuit topological interpretations of several numerical phenomena is possible Motivation mequations lttodelreduction circuitsymhesis Exampts Sutrnnaryan tlook References Contents 9 Circuit synthesis Find incidence and component matrices such that circuit transition behaviour coincides with that of Find incidence and component matrices such that circuit transition behaviour coincides with that of Problem with known results in circuit synthesis 0 difficult to manage numerically sometimes only proofs of existence u often based on ordinary differential equations Numerical algorithm for synthesis R Bring the system into a form 7 E11 0 7 A11 A12 7 T7 31 0 Li iii oiB C io i 0 E22 4172 with E11 E22 7A symmetric positive definite circuit synthesis Numerical algorithm for synthesis R Bring the system into a form 7 E11 0 7 A11 A12 7 T7 31 O Lie oiHiio i with E11 E22 7A symmetric positive definite Second step Bring the system to the above form with moreover E11Z 052 7 2 11 E22 diagZ o 0 E1 7 Z RNRAZ A 2 127ZL lrrh nT 72W 71 E240 0 IN Moiivaxion equations Modell edumion 39cunsymhesis Example Suurnnaryan References Contents Example Example Circuit RC chain NEC L 1 1 S1 I T I I 0 63 capacitances 0 63 resistances 0 64 nodes 0 one voltage source Descriptor system with state space dimension 65 Motivatien Circuit equations Modelr union Circuit synthesis Example Summaw and Outlook References Circuit RC chain Passivitypreserving model reduction yields system with state space Fvequency vespunses 3 Absulule enuv m 7M Hide 7 7 39PR ET u 025 u 02 g n ma g 10 E u m u 005 10 105 1D 1D in 1D 1D in m 1 0 0 00024 00050 0 00077 00050 00327 0 BCT 00181 0 0 1 01583 Motivation requarions Model reduction Ci rcu Example Summary and Outlook Relerences Synthesis of reduced model yields circuit with o 2 capacitances o 1 inductance o 3 resistances o 2 transformer with one initial port and three terminal ports 0 1 voltage source 0 4 nodes Incidence matrices Ac AR An soda 1 O O O and element matrices 030 324 O 712 0 l U U U 15 83 U 9 2 o o o o o o 1 VAT li 0 WM 0 o 1 1 075 0 10quot L57410 9 T 018 2113 75 31 Approximating circuit uvt Moi axiom quations Modell edu 39 Exampm Summaryandoutlook Refer Contents a Summary and Outlook J equalions V 4 L Example Summaryandomiook Reterulc Summary 0 Circuit equations as passive descriptor systems 9 Passivity preserving model reduction 0 Circuit synthesis of passive descriptor systems Motivation Cilcuhequations Modelreuucuon Cucqu IS Example Summaryandouuook Relerences Outlook open problems 0 Lur e solver for nonstrictly passive systems with E Sachs T Stykel o Inversefree balancing for fractional Gramians X X1X2quot Y Y1 Y1 0 Integration of the method in industial circuit simulation packages with NEC Europe 0 Further exploitation of structure of circuit equations with T Stykel 0 Consideration of nonlinear effects 0 Inclusion of distributed components transmission lines Maxwell etc with B Jacob TU Delft g You can download my articles from WWW4math4tLliberlln4d9rels AndeV73 BDO Anderson and S Vongpanitlerd NetworkAnalysis and Synthesis Prentice Hall Englewood Cliffs NJ 1973 GuoLan98 CH Guo and P Lancaster Analysis and modification of Newton39s method for algebraic Riccati equations Mathematics of Computation 67 1998 pp 1089 1105 JR Phillips L Daniel and LM Silveira Guaranteed sive balancing transformations for model order reduction IEEE Trans ComputerAided Design 22 2003 pp 1027 1041 Rei08 T Reis Circuit synthesis of passive descriptor systems a modified nodal approach International Journal of Circuit Theory and Applications to appear 2008 ReiSty08a T Reis and T Stykel Positive real and bounded real balancing for model reduction of descriptor systems Preprint 252008 Institut fur Mathematik TU Berlin submitted 2008 ReiSty08b T Reis and T Stykel lanced Truncation for Electrical Circuits will be finished soon 200
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