Advanced Computational Methods for VLSI Systems. Lecture 3 Basic circuit analysis methods. Zhuo Feng
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1 Adanced Compuaonal Meho for VLSI Sysems Lecure 3 Basc crcu analyss meho Zhuo Fen 3.
2 Read Chap. 4 c fle 0 dc nonlnear soln. open C s ; shor L s nser TR models for L s and C s dc nonlnear soln. for equalen CKT? fnal Done 3.
3 3.3 One-sep neraon appromaon d d C C d C ( ( ( ( ( ( ( ( d Forward Euler (FE Bacward Euler (BE Trapezodal (TR
4 FE s eplc, and no nonlnear eraons are requred Eremely dffcul o use n pracce ( Forward Euler ( d ( 3.4
5 BE s mplc Much more robus han FE Can also mae unsable responses appear sable ( Bacward Euler ( d ( 3.5
6 TR s mplc oo Wors smlarly o BE Incurs less error ( Trapezodal ( d ( ( 3.6
7 FE Capacor Companon Model C C d d ( d ( ( ( ( ( C 3.7
8 BE Capacor Companon Model Theenn C C d d ( d ( ( V eq ( R eq C 3.8
9 BE Capacor Companon Model Noron ( I eq C ( ( G eq C 3.9
10 TR Capacor Companon Model Theenn C C d d ( d ( ( ( V eq ( ( C R eq C 3.0
11 TR Capacor Companon Model Noron ( I eq C ( ( ( G eq C 3.
12 TR Inducor Companon Model Noron L Ld d ( ( ( d L ( d ( ( ( I eq L ( ( ( G eq L 3.
13 TR Inducor Companon Model Theenn ( V eq R eq L ( ( L 3.3
14 Wha error s ncurred n each appromaon? Taylor s seres ( ( ( ( ( Assumn ha he soluon s accurae Frs erms of seres represen F.E. FE 3 ( ( 6 3.4
15 For a coneren seres, usn a dfference appromaon for he nd order derae: Bacward Euler error Rewre he Taylor seres Reorder ( ( C FE ( ( ( ( ( ( ( ( ( ( 3.5
16 ( ( ( BE C Same manude as FE bu oppose sn TR error Dern TR error s a b more comple Bu for a smple RC crcu we can show TR RC ( ( Error s proporonal o 3.6
17 We can use a dfferen approach o dere he local runcaon error (LTE For hs smple RC crcu he eac soluon s: c ( ( 0 ( RC RC c0e e ( d RC Zero-npu Zero-sae R C 0 c 3.7
18 Consder a semen of ramp npu V n n V n Eac soluon RC, ( (, V RC e ( Vn RC c n c n n n 3.8
19 Epand e n RC n RC! n RC 3! n RC 3 FE s equalen o reann only he frs wo erms For BE we rearrane he eac soluon RC c, n ( ( c, n Vn RC e ( Vn RC n c, n V n RC c, n Vn RC RCe n 3.9
20 BE s equalen o reann he frs wo erms n he epanson: e 3 n RC n n n RC! RC 3! RC For TR we rearrane he eac soluon o he form RC RC V RC e V RC e c, n n n c, n n n TR s equalen o eepn he frs wo erms n he epansons for boh! 3.0
21 We hae looed a he error ncurred a one sep of he numercal neraon Assume he soluon s accurae Ths s a measure of errors ncurred locally Local runcaon error (LTE We also need o loo how errors accumulae lobally Decay --- sable Grow --- non-sable Need o noe a sably creron 3.
22 Tae anoher loo a he eac soluon we dered for he smple RC crcu c RC, n( ( c, n Vn RC e ( Vn RC n ransen soluon parcular soluon npu parcular oal ransen 3.
23 We can arrane he eac soluon o hae c, n RC c n e c, n c, n, n < The dfference beween he oal soluon and he seady-sae soluon s decreasn oer he me, We noe c n c, n c, n c n as a sably creron, 3.3
24 For FE, sably mples ha e n RC n RC n RC RC n 3.4
25 For BE, we hae e n RC n RC e n RC n RC n RC True as lon as 0 n 3.5
26 For TR, we hae e n RC e e n RC n RC n RC n RC n RC n RC Same as BE, rue as lon as 0 n 3.6
27 FE, BE and TR are one-sep neraon meho People hae also red o use hh order neraons o mproe he accuracy Runa-Kua meho Gear s mehod For dal crcus, hese hh order meho may no be que useful 3.7
28 Read: Chap Las me we aled abou lnear sysem soluon and he sparse mar daa srucure For a nonlnear crcu, we need o sole a se of nonlnear equaons Newon Raphson mehod Nonlnear DC analyss Nonlnear ransen analyss Harmonc balance We frs al abou he nonlnear DC analyss Oher ypes of nonlnear analyses can be conered o equalen nonlnear DC problems 3.8
29 Eample: Nonlnear Transen Flow c fle 0 dc nonlnear soln. open C s ; shor L s nser TR models for L s and C s dc nonlnear soln. for equalen CKT? fnal Done 3.9
30 Assume we are soln a scalar nonlnear equaon p( 0 Taylor s heorem For a coneren seres - - small enouh p ( p( p( Where: p ( Imporan: Requres esence of p( o sole a Newon mehod 3.30
31 3.3 Newon Raphson: 0 ( ( ( p p p ( ( p p Soln ( ( p p For suffcenly small wors Whou esence of p ( can oscllae
32 p( D eample 0 p( p( Ierae unl conerence Mos realsc problems requre dampn 3.3
33 N-R for N-Dmensonal Problem P( 0 p p ( 0 ( 0 P P( P( Jacoban of 3.33
34 3.34 p p p P J 0 ( ( J P P Frs wo erms of Taylor Seres:
35 3.35 Epanded: 0 ( 0 ( 0 ( p p p p p p p p p N N N
36 3.36 N-R Ieraons Impraccal o ner J : ( P J ( P J J
37 For lare CKTs he lnear sole can realy mpac he run me (mllons of nodes For smaller cs, buldn he Jacoban for comple dece equaons always domnaes he run me How we buld he Jacoban s mporan We don hae a symbolc ( parally dfferenae! P ha we can Dece equaons end o be sem-emprcal and que messy! Paral deraes are een messer! 3.37
38 Eample : d R 3 R Nonlnear dc eample 4 d f d V T ( I ( e d s 4 3 f ( R R
39 Lnearzn he dode model n erms of frs wo Taylor seres erms: f ( d df d d d d f ( d f ( d df d d ( d d 3.39
40 Formulae nodal equaons o sole for Ths s he lnearzed soluon for a nonlnear eraon f ( d G EQ Load lne for remander of CKT I EQ d d 3.40
41 Correspon o a lnearzed equalen CKT model for dode: d ( d d d d d d d I EQ d d d d d ( d G EQ d d d d Small snal dode conducance Paral derae sored symbolcally for each dode model G EQ d I EQ 3.4
42 Lnearzed Equalen CKT I EQ G EQ Sole o oban, and so on 3.4
43 To fnd he DC soluon, open C s and shor L s No dynamcs DC soluon for a lnear crcu s easy One lnear sysem soluon Fndn a ood nal uess mh be nonral for a nonlnear crcu Specal rcs are needed for fndn he nal uess and adn he nonlnear DC soluon conerence Some form of connuaon meho 3.43
44 Wha nal node olaes should we pc for sarn our nonlnear DC eraons? + _ + _ + _ 3.44
45 Emulan he power-on process Replace he DC power supply olae source by a lon rse me ramp npu Se all he nally node olaes as zero Perform a ransen analyss unl he ramp npu reaches he fnal power supply olae Use he soluon a he las me sep us he DC soluon of he crcu 3.45
46 In SPICE a dfferen approach s used GMIN seppn Aachn a small dummy conducance beween each node and he round o ad he conerence mn mn + _ + _ + _ mn 3.46
47 GMIN Seppn Gmn = Inal alue; whle Gmn!= 0 { coun = 0; conered = false; whle (! conered { f (coun > lm fal; Updae he lnear problem; Sole he crcu; conered = conerechec(; coun ++; } reduce Gmn; } 3.47
48 Read Secons Recap : nonlnear DC analyss Sore dece equaons and her paral deraes w.r.. branch olaes for effcen N-R procedure: Inser lnearzed models no MNA formulaon (frs order Taylor seres a operan pon Sole he lnear c o complee one N-R eraon Use he soluon as operan p. for ne lnearzaon sep 3.48
49 3.49 Dode equaons and companon model for N-R d ( T V d e I s d d VT T s d d e V I ( d d d d d d ( d d d d d d Sored Model Eqns.
50 Dodes are modeled by Noron equalen companon models I eq K d d d ( d Sole o oban and so on G eq d d Some sor of olae lmn scheme s requred o mae N-R eraons robus 3.50
51 Recap: lnear ransen analyss Replace each of C s and L s by a companon model numercal neraon Forward Euler, Bacward Euler, Trapezodal Noron or Theenn models Sole he equalen lnear crcu a he curren me sep Updae all he companon models and moe o he ne me sep 3.5
52 Eample: BE capacor companon models Theenn C C d d ( d ( ( V eq ( R eq C 3.5
53 Noron ( I eq C ( ( G eq C 3.53
54 We combne he aboe wo n nonlnear ransen analyss N-R eraons ( Numercal neraon n n 3.54
55 Sar from some nal condon a me 0 Moe o he ne me sep by replacn all he C s & L s by a companon model Sole he equalen nonlnear DC problem a he new me pon Use he soluon a he preous me sep as he nal uess for N-R Ierae ll conerence Repea ll reachn he endn me Nonlnear dynamc elemens need o be handled more carefully Chare conseraon more on hs laer 3.55
56 Wha abou nonlnear elemens wh more han ermnals? 3 Nonlnear equaons:,, ( ( 3 Por Equaons Once aan, model by frs erms of Taylor seres 3.56
57 3.57, (, ( 3 3 3, (, ( K
58 Termnal MOSFET Samp: Consder he curren conrbuon a each node Noe ha we mus ranslae por olaes o node olaes a b c, ( c b c a c b a, (
59 Termnal MOSFET Samp: Mos solers are se up o sole for nsead of a b c a b c, (, (, (, ( a b c c b a
60 MOSFETs (3 ermnal d s s Trode Reon ( s TH s TH C o W L s TH
61 Sauraon Reon ( s TH s TH Cuoff Reon 0 s TH 3.6
62 3.6 dc por equaons:, ( s d 0, ( s d s ( (, ( s s s s d d d
63 3.63 s TH s V m s G s m G
64 Equalen c model for N-R m s G G s m s G S s s s 3.64
65 Could also buld models o sole for and drecly s Sampn n erms for all of hese -ermnal elemens s equalen o applyn MOSFET samp Noe he lare off-daonal erms ha are creaed by m ' s 3.65
66 Smple Eample I m EQ s m G G s V IN R R M 0 R 3 V DD R R 3 V DD V IN R m I EQ G 3.66
67 Now formulae he nodal equaons for hs lnearzed equalen c R R 3 V DD V IN R m I EQ G 3.67
68 , I G m and EQ alues chane a each N-R eraon Dampn meho conrol he N-R conerence More of a problem for BJTS In eneral, we would nclude he 4-h ermnal of he MOSFET (body effec 3.68
69 We use dc nonlnear alorhms fnd soluon a =0 AND for all mepons Enery sorae elemens are properly consdered by numercal neraon C How abou nonlnear capacors? 3.69
70 I EQ GEQ and are nonlnear fcs. of Lnearze and sole a N-R Unforunaely no all caps are handled hs easly MOSFET C s are more dffcul because he capacance n a nonlnear fc. of mulple olaes 3.70
71 G C GS C GD S D B B B The nonlnear capacance formulas chane as a fc of he reon of operaon 3.7
72 Cu-Off: V GS V TH L n p n C GB C OX W L (effece W,L C C GS GD C C OX OX W L W L Oerlap componens 3.7
73 Wh an nduced channel C GB 0 Gae o channel cap s rbued beween ae and source & ae and dran C OX W L C C GS GD C C OX OX WL WL Channel componens 3.73
74 Lnear Reon : VGS VTH VGD VTH L n Channel p n 3.74
75 Lnear Reon C OX WL C OX WL When dece eners sauraon, here s no loner a channel connecon o dran Model for hs was dered emprcally 3 C OX WL 3 C OX WL 3.75
76 Bu we now ha models mus hae connuy across reons for N-R Consern chare s rcy snce C s are fcs of all olaes d C d can be confusn! Bac o Bascs: C s C d C C q f, q f, Use q as he sae arable 3.76
77 Conenonal NL dece models = f(, s, sb parameers Accurae ealuaon can be ery epense NR requres full ealuaon of models Deraes requred for Jacoban Epense o ealuae -- can consume 50-80% of compuaon me SC-based approach No deraes o re-oranze appromae Jacoban Wll wor wh able models, measured daa Reduced model ealuaon me, bu s based on: Slower conerence (dependn on he J appro Mar updae requred for aryn meseps Selecon of represenae lnear model n J appro 3.77
78 Newon-Raphson Good conerence properes Relable when mplemened wh a sep-lmn mechansm (damped Requres eplc dfferenaon n-d case: Jacoban Mar sampn (updae and refacorzaon
79 Successe Chord J( represens a fed raden (Jacoban No eplc dfferenaon Relable when mplemened wh a sep-lmn mechansm (damped n-d case: Snle Jacoban facorzaon mproemen by oponal updaes
80 NR has aryn R, I SC has fed R Lnearzed newor chanes for each NR sep,.e. J( n Snle lnearzed newor for SC seps, J appro Newon-Raphson Successe Chord 3.80
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