Semistate Theory and Design of Analog VLSI Circuits

Transcription

1 Semstate Theory and Desgn of Analog VLSI Crcuts Roert W. Newcom Electrcal and Computer Engneerng Department Unersty of Maryland, College Park, MD 74 USA URL: emal:

2 Tale of Contents. Ttle. Tale of Contents 3. Descrng Equatons I 4. Descrng Equatons II 5. Descrng Equatons III 6. Descrng Equatons IV 7. Settng Up Equatons I 8. Settng Up Equatons II 9. Settng Up Equatons III. Settng Up Equatons IV. Settng Up Equatons Eample I. Settng Up Equatons Eample II 3. Settng Up Equatons Eample III 4. Settng Up Equatons Eample IV 5. Semstate for LTI Crcuts I 6. Semstate for LTI Crcuts II 7. Semstate Propertes - I 8. Semstate Propertes - II 9. Semstate Desgn - I. Semstate Desgn - II. Semstate Desgn III. Semstate Desgn - IV 3. Semstate Desgn - V 4. Semstate Desgn - VI 5. Semstate Desgn - VII 6. Semstate Desgn VIII 7. Semstate Desgn - IX 8. Basc VLSI Components 9. Dfferental Par - Crcut 3. Dfferental Par - Layout 3. LTI Semstate Canoncal Form - I 3. LTI Semstate Canoncal Form - II 33. LTI Semstate Canoncal Form - III 34. References

3 Descrng Equatons - I General Nonlnear Tme-Varyng Crcut F(,,u,t) G(,y,u,t) nternal descrpton ector, k-ector tme derate u nput, m-ector y output, n-ector Dffculty: too cumersome to use for most purposes of analyss or synthess.

4 Descrng Equatons - II State Varale Equatons F(,u,t) y G(,t) nternal state, k-ector tme derate u nput, m-ector y output, n-ector Adantage: coered y years of mathematcal theores Dffculty: must augment G separately to nclude resstors, dfferentators, etc.. G(,t) G(,u,u, u,...,t)

5 Descrng Equatons - III Semstate Equatons State Equatons Canoncal Form E Α (,t) Bu y C B, C, E constant matrces nternal state, k-ector tme derate u nput, m-ector y output, n-ector Ths s the form needed for VLSI where A(,t)A() s usually nonlnear.

6 Descrng Equatons - IV Lnear Tme-Inarant (LTI) Crcuts E A Bu y C ( Du) A, B, C, D, E constant matrces sem-state, k-ector tme derate u nput, m-ector y output, n-ector If E k reduces to state arale case. Most useful tme-doman descrpton for desgn of lnear crcuts. We wll assume all quanttes real.

7 Settng Up Equatons - I Use graph theory: numer tree ranches frst and lnks last Assume only one separate part y attachng common grounds ndependent tree ranches tree & # of t ndependent lnk ranches lnks & # of ranches ranches & # of te-set matc N ranch KVL,equaton for each, cut-set matr M t KCL,equaton for each t ranch, T T C C t, t

8 Settng Up Equatons - II ; T T T T -M N sources, are ndependent and t where graph ests As a N M t t Power_n -M t, M t M, t M t t T T T T T C t

9 Settng Up Equatons - III Assume, for conenence of seeng the theory, and y aalale equalences usually usng gyrators, that all dynamcs s n capactors and all sources are current sources. For a lnear tme-narant crcut we can also assume a ranch y ranch admttance matr ests. Y g g

10 Settng Up Equatons - IV Generc Graph Element J E f( gs, d ;β) ds K β P W L f gs -V th ds β(( gs V ) )( λ ) th ds ds ds f gs -V th ds β( gs V )( λ ) f gs -V th ds th ds f gs -V ds th

11 Settng Up Equatons Eample - I Current Mrror Graph t C 5 y ; n I u

12 Settng Up Equatons Eample - II KCL cut sets for nodes I & II C KVL te sets for lnks 3,4, T t M, t M t T

13 Settng Up Equatons Eample - III Dece Equatons u GVdd 3 M dd GV n I J ) ;β, f( ) ;β, f( G C T dd GV n -I J J;,

14 Settng Up Equatons Eample - IV Fnal Semstate Equatons y u -f(,,β) -f(,,β) GVdd - -G - - C Common case: β) f(,; β β ;β) f(, ) ;β, f( y u ) ;β, f( C dd GV -G ) ;β, f( 5 u y β β C u L L W W u β β y C Reducng

15 Semstate for LTI Crcuts - I C y Bu A E Eamples: Resste crcut on admttance ass: YCA - B Admttance matr, u, y: Y(s)C(sE-A) - B We wll assume a regular system.e. nerse ests for se-a Derate: - y u - - su or y s det as det(se-a) s B A) C(sE

16 Semstate for LTI Crcuts - II Addton of admttances: C C y u B B A A s E E, y y y, u u u )-B -A (se C (s) Y ; B ) A (se C (s) Y

17 Semstate Propertes - I Inde of a matr: smallest nonnegate for whch ) rank(a ) rank(a Eamples: nde 3 nde nde Nonsngular nde

18 Semstate Propertes - II nde4 nde, Physcal meanng: Sum of (lock nde-) # of dfferentators needed lock ndces (3,) nde 3;

19 Semstate Desgn - I Y(s) Y Y (Y ( s) Y ) Y L Compare wth Y(s) C(sEA) B Therefore dentfy se Y L y Y (s) Y Y u In order to realze Y L wth capactors, for VLSI, transform E to symmetrc poste semdefnte

20 Semstate Desgn - II Transformaton: multply y P and replace y Q to replace E y PEQ where C ap C ap C ap k c drect sum, T andc ranke

21 Semstate Desgn - III CQ PB PAQ ]s c k [C ap PEQs Q n A B C P n PAQ PQ CQ couplng Y c k sc ap Load Y Brng C ap to dagonal poste defnte form (always possle) to synthesze y uncoupled poste capactors. Synthesze Y couplng y dfferental pars. Note that Y couplng may e acte.

22 Semstate Desgn - IV Transfer functon synthess: Transform to Y and use gyrators to get u, y Case of oltage transfer functon: Identfy: 3 -g out -gy, out, u, don t care

23 Semstate Desgn - V don't care, A B C couplng Y 3 A B C u A B C se gy (s) L Y out g 3 Therefore, synthesze Y couplng y VCCSs dfferental pars and load n Y L (s) capactors and opens.

24 Semstate Desgn - VI Agan transform wth P & Q: YL(s) speq sc ap kc Y couplng PB CQ PAQ Choose the don t care so the couplng admttance s as lossless (skew symmetrc as possle and PAQ as passe as possle. Eentually scale for VLSI. Y couplng PB (CQ) T (PB) CQ T PAQ

25 Semstate Desgn - VII Eample: s s 6 3s 6 3s s s 3s3 Realze as the sum of three admttances For 3s: For 6: For : s s 6 3s 3 a 6 y a u a s a y u s 6 c y c u c s c

26 Semstate Desgn - VIII y u s To get E dagonal we permute the 3 rd and 4 th columns usng 5,Q P Fllng n the don t care entres to otan skew-lke symmetry couplng Y

27 Semstate Desgn - IX Ths s realzed y a seen port of dfferental pars, some ack to ack as gyrators. The second port has a gyrator, of gyrator conductance, to conert to -g out. The last 5 ports are loaded wth 3 unt capactors and two open crcuts. Note that the thrd capactor could hae een placed n parallel wth the nput ut that posslty s outsde of ths desgn method (though not outsde of semstate theory) snce no couplng Y would est.

28 Dfferental Par Basc VLSI Components Symol VCCS Spce G Y G PMOS & NMOS Capactors

29 Response Dfferental Par - Crcut

30 Dfferental Par - Layout [Ne3]

31 LTI Semstate Canoncal Form - I C C y u B B σ k A l N σ Where N l s nlpotent (or asent f E s nonsngular). Proof: Rather messy (n Gantmacher [GA]) ; A ) nl E α(e ns ) nl E α)(e ns (s TAT wth A A ) nl E s(e ns A)T T(sE ) nl E (E ns A ) nl E α(e ns α) (s k A) (se then A ) nl E α(e ns Multply y nerse of nonsngular. s such that Choose real A ) nl E α(e ns α The standard canoncal form s

32 LTI Semstate Canoncal Form - II Wth a new T rng k ) nl F ns α(f nl F ns F s or se-a nl F F ns ) nl E (E ns A ) nl E α(e ns Multply y the nerse of k-σ nl αf F ns Sends se-a to the desred form ) nl F σ nl αf ( ]) F ns [ (s σ σ α σ k s

33 LTI Semstate Canoncal Form - III Use n desgn: (t) c (t) y(t) t [A (τ ) B d(n (t)) l B u(t) dt C (t) C (t) u(τ )]dτ () For synthess replace the mddle term y ts transformed such that the derate term s dagonal s( ) A c kcc B u Synthess can now take place y the use of cc unt capactors fed y dfferental pars for A, A, B and B wth uoltage, ycurrent.

34 References [CA] S. L. Campell, Sngular Systems of Dfferental Equatons, Ptman Adanced Pulshng Program, San Francsco, 98. See p. 44 for Drazn nerse soluton. [CA] S. L. Campell, Sngular Systems of Dfferental Equatons II, Ptman Adanced Pulshng Program, San Francsco, 98. See p. 6 for the standard canoncal form. [GA] F. R. Gantmacher, The Theory of Matrces, olume two Chelsea Pulshng Company, New York, 959. See p. 8 for the deraton of the standard canoncal form [Ne] R. W. Newcom and B. Dzurla, Some Crcuts and Systems Applcatons of Semstate Theory, Crcuts Systems and Sgnal Processng, Vol. 8, No. 3, 989, pp [Ne] R. W. Newcom, Semstate Desgn Theory, Crcuts Systems and Sgnal Processng, Vol., No., 98, pp [Ne3]