A NEW GENERALISED HYBRID METHOD FOR NONLINEAR ANALOG CIRCUIT ANALYSIS

Transcription

1 A NEW GENERALISED HYBRID METHOD FOR NONLINEAR ANALOG CIRCUIT ANALYSIS MIHAI IORDACHE LUCIA DUMITRIU DRAGOŞ NICULAE Ky word: Nonlnar analog crcut Hybrd mtod Trannt baour Frqunc doman approac. T papr prnt a nw ron of t gnrald ybrd mtod for computng of t nonlnar analog crcut rpon. T mtod calld ybrd analy allow a mxtur of bot currnt and oltag-controlld rtor oltag-controlld (.c.) nonlnar capactor currnt-controlld (c.c.) nonlnar nductor lnar capactor nductor rtor ndpndnt ourc and all four typ of lnar controlld ourc. Hybrd mtod rmarabl by t grat ffcncy and gnralty and t ry uful bot n t trannt baour and n frquncy-doman approac bad on armonc balanc and lat quar approxmaton. An llutrat xampl prntd.. INTRODUCTION In ordr to rduc t computng tm and t mmory t ncary to parat t lnar crcut part from t nonlnar on. T paraton allow t calculaton of to trm from t crcut quaton dpndng of t paramtr of t lnar crcut lmnt only onc at t bgnnng of t computng proc. By rplacng ac capactor and nductor by a dcrt rt modl aocatd wt a prlctd mplct numrcal ntgraton (a: bacward Eulr algortm trapzodal algortm and Gar cond-ordr algortm) t trannt analy of a nonlnar dynamc crcut can b rducd to t dc analy of a qunc of nonlnar rt crcut [ 8 ]. Among t many dtnct mtod for formulatng crcut quaton t m- port ybrd-analy approac [ 2 and 6]. T mtod mor gnral and mor ffcnt wn t undr-condraton crcut contan a larg prcntag of lnar crcut lmnt and controlld ourc. Anotr adantag of t ybrd-analy approac tat t allow t nonlnart to b tr oltag-controlld (c..) or currnt-controlld (c.c.) a flxblty not ard by nodal-analy mtod. T bac ploopy to form an m-port Ĉ from t gn crcut by xtractng an approprat t of two-trmnal lmnt o tat t rultant m-port contan only Poltnca Unrty of Bucart E-mal: ordac@lt.pub.ro R. Roum. Sc. Tcn. Élctrotcn. t Énrg p Bucart 2008

2 394 Ma Iordac Luca Dumtru Dragoş Ncula 2 lnar capactor nductor rtor and lnar controlld ourc. T mtod ry uful n frquncy-doman approac bad on armonc balanc and lat quar approxmaton [4 ]. Our obct n t papr to dlop a gnral algortm for formulatng t ybrd quaton of a ry larg cla of nonlnar crcut. T mtod calld ybrd analy - allow a mxtur of bot currnt and oltag-controlld rtor oltag-controlld (.c.) nonlnar capactor currnt-controlld (c.c.) nonlnar nductor lnar capactor nductor rtor ndpndnt ourc and all four typ of lnar controlld ourc. 2. DESCRIPTION OF THE METHOD To aod dalng wt crtan typ of crcut wo ybrd quaton tr do not xt or ar patologcal n t n tat ybrd quaton oluton dpnd on t prc alu of om lmnt paramtr w all aum tat our crcut mt t followng rqurmnt:. Contncy aumpton: a) Tr do not xt any loop mad up of only ndpndnt and/or controlld oltag ourc (E loop); b) Tr do not xt any cutt mad up of only ndpndnt and/or controlld currnt ourc (J cutt). 2. Aumpton on controllng arabl: t controlld ourc can b dpndd only t currnt (t oltag) of t c.c. (c..) nonlnar crcut lmnt and of lnar rtor (ncluly t lnar rtor from t companon modl corrpondng to t lnar dynamc lmnt). 3. Normal-tr aumpton: W coo a pcal tr calld t normal tr (NT) wo lmnt ar lctd accordng to t followng prorty: a) all ndpndnt and controlld oltag ourc; b) all c.. nonlnar crcut lmnt (capactor and/or rtor); c) a many lnar rtor (ncluly t lnar rtor from t companon modl corrpondng to t lnar dynamc lmnt). NT do not contan any ndpndnt and controlld currnt ourc and any c.c. nonlnar lmnt (nductor and/or rtor). Lt C b a crcut tat atf t tandng aumpton and lt T b a normal tr and L t corrpondng cotr. Our frt ta to form a lnar m- port Ĉ wt m n n n u n (wr for xampl n u (n ) t c.. (c.c.) nonlnar lmnt numbr) obtand by xtractng from C all ndpndnt ourc all c.. nonlnar lmnt (capactor and rtor) and all c.c. nonlnar lmnt (nductor and rtor) a own n Fg. a. In w of our procdur for lctng t NT all xtractd lmnt on t lft of Ĉ ar tr branc and trfor conttut a part of NT and all lmnt on t rgt of Ĉ ar cotr

3 3 Gnrald ybrd mtod for nonlnar analog crcut analy 395 branc (ln) and blong to t cotr L. T rmanng lmnt n t m-port Ĉ cont only of lnar capactor nductor (magntc coupld or not) rtor and lnar controlld ourc. Subttutng all nonlnar lmnt from t lft d n Fg. a by dal oltag ourc and all nonlnar lmnt from t rgt d by dal currnt ourc and rplacng t lnar capactor and nductor by tr rt dcrt crcut modl aocatd wt a gn ntgraton algortm (for xampl t bacward Eulr algortm) w obtan t lnar and tm-narant crcut n Fg. b. Fg. a) T lnar m-port Ĉ cratd by xtractng all ndpndnt ourc and all nonlnar lmnt; b) lnar m-port Ĉ wt t tr oltag port apparng on t lft d and t cotr currnt port apparng on t rgt d; Applyng t uprpoton torm to t lnar m-port Ĉ n Fg. b for t tm momnt t t ( bng t tm tp) wn all lnar capactor and all lnar nductor ar rplacd by dcrt rt modl aocatd to bacward Eulr algortm and at t ( t traton of t Nwton-Rapon algortm t rult:. C L C L C L R A B G R A B G R A B G (

4 396 Ma Iordac Luca Dumtru Dragoş Ncula 4 B u A u rprnt t currnt (oltag) tranfr coffcnt matrx of t tr-branc (ln).c. (c.c.) nonlnar lmnt n rpct of t ln (tr-branc) ( ( c.c. (.c.) nonlnar lmnt; ( ( ( ) t oltag (currnt) ctor of t.c. (c.c.) tr-branc (ln) nonlnar lmnt from t tm momnt t and t ( t ( ) ( ) traton; L ( C ) rprnt t oltag (currnt) ctor of t of dal ndpndnt oltag (currnt) ourc from t companon cm of lnar nductor (capactor) at t tm momnt t (or at prou tm tp to t ). If w dnott by: In ( ( ( ) G B X ( ( ; H ; x( ( ; ( A R ( G ( ) B G L B ( ) C L S ; S LC ( ) A R A L R C C t quaton ( bcom ( ( ( ( ) X ( Hx( S S LC (3) ( wr: H crcut ybrd matrx; S rprnt t ourc ctor corrpondng to t ndpndnt oltag and currnt ourc from t tm t and ( ) S LC t ourc ctor corrpondng to t companon cm of t lnar nductor and capactor at t prou tm tp to t. T nonlnar rtor caractrtc approxmatd by pcw lnar contnuou cur a for t tm momnt t and t ( t traton t followng xpron: ( ( ( ( Ru ( Gdu ( ( ) ) uru ( Ru ( ( ) ) (4) for t.c. nonlnar rtor and ( ( ( ( u R ( Rd ( ( ) ) R ( R ( ( ) ) (5) for t c.c. nonlnar rtor. Accordng to t quaton (4) ((5)) ac.c. (c.c.) nonlnar rtor for any arbtrary gmnt ( ) ( ) (for t tm momnt t and t () t traton) can b ubttutd by t qualnt crcut own n Fg. 2a (Fg. 2b). (2a) (2b)

5 5 Gnrald ybrd mtod for nonlnar analog crcut analy 397 Fg. 2 Equalnt crcut for t pcw lnar nonlnar rtor. T currnt xpron of a.c. nonlnar capactor wn t caractrtc approxmatd by pcw lnar contnuou cur and for t tm momnt t and t (-t traton (ung t bacward Eulr ntgraton algortm) a t followng xpron: ( ( ( ( ) ˆ C GdC C ( C C ( ( C ( ) ( ) q ( ) ( ) d C ˆ q (6) C GdC C C. Accordng to t quaton (6) ac.c. nonlnar capactor can b ubttutd by a dcrt rt modl aocatd wt: bacward Eulr algortm own n Fg. 3. T oltag xpron of a c.c. nonlnar nductor wn t caractrtc approxmatd by pcw lnar contnuou cur and for t tm momnt t and t (t traton (ung t bacward Eulr ntgraton algortm) a t followng xpron: Fg. 3 Dcrt rt modl aocatd wt: bacward Eulr algortm for a.c. nonlnar capactor.

6 398 Ma Iordac Luca Dumtru Dragoş Ncula 6 ( ( ( u ( ) ˆ L RdL L L L ( ( L ( ) ( ) d Φ L ( ) RdL ˆ L ( ) L Φ Equaton (7) lad to t dcrt rt modl aocatd wt: bacward Eulr algortm own n Fg. 4. L. (7) Fg. 4 Dcrt rt modl aocatd wt: bacward Eulr algortm for a c.c. nonlnar nductor. Introducng nto quaton ( t lnar pcw caractrtc of t nonlnar crcut lmnt w obtan t crcut quaton from t tm momnt t and t ( t traton ( Gd ( ( ) ) G B ( ) ( ( ) ( ( ( S S LC ( ) ( ). (8) A Rd ˆ R Equaton (8) conttut m m m 2 ndpndnt quaton n m unnown oltag oltag ctor ( ( and m 2 unnown currnt currnt ctor ( ( ) and ar calld t ybrd quaton of t nonlnar crcut C. W obr tat nc t ybrd ubmatrc G B A and R and t two ourc ctor ( ( ) S and S LC (rlaton (2 b)) ar fxd ( only Gd ( ( ) ) Rd ( ( ) ) ( ( ) ). and ( ( ˆ ) ) nd b cangd n ac traton. Trfor t Jacoban matrx n Eq. (8) can b obtand mpl from t lop of t approprat gmnt of t nonlnar cur. Kpng a ymbol only t paramtr aocatd to t nonlnar crcut lmnt t Nwton- Rapon algortm bcom ry ffcnt. Structur of t ybrd quaton (0)

7 7 Gnrald ybrd mtod for nonlnar analog crcut analy 399 adquat to ol t nonlnar crcut by t lctrcal macn atng and/or coolng ytm modlld. 3. EXAMPLE Lt b t nonlnar crcut own n Fg. 5a. (a) (b) Fg. 5 a) Dagram crcut; b) lnar m-port Ĉ. T algortm of t ybrd partal ymbolc analy for t nonlnar crcut n Fg. 5 a cont n t followng tp:. Accordng to t aumpton from Scton II w gnrat t normal tr. Tr branc rprntd n Fg 5 a by dad ln. 2. W ubttut at t tm momnt t and t (-t traton all c.. nonlnar lmnt by dal oltag ourc and all c.c. nonlnar lmnt by dal currnt ourc and rplacng t lnar capactor and nductor by tr rt dcrt crcut modl aocatd wt a gn ntgraton algortm (for xampl t bacward Eulr algortm) w obtan t lnar and tm-narant crcut n Fg. 5b. 3. For t numrcal alu of t lnar lmnt paramtr: L 7 µh C 6 2pF R 6 2Ω C 7 4pF R 7 Ω G 5_6 0.00S a 8_ R C6 500Ω R L7 Ω and R C7 250Ω t analzd by paral ymbolc modfd nodal mtod [0 2 4] t lnar m-port n Fg. 5 b. In t way w obtan t

8 400 Ma Iordac Luca Dumtru Dragoş Ncula 8 ybrd quaton (. Runnng SYMNA program [0] w obtan t followng ybrd quaton: ( ( u ( ) ( ( 2 ( u 2 ( ( ( u 3 ( 3 ( ( ) L7 ( ) ( ( ) 4 C6 ( ( ) 9 ( ) C7 ( 4. Introducng nto quaton (9) t lnar pcw caractrtc of t nonlnar crcut lmnt w obtan t crcut quaton from t tm momnt t and t ( t traton: ( G ( ) du ( GdCu2 ( ( ) ) ( ( ) RdL3 ( u L7 ( 2 ( u C6 ( (0) ( C 7 ( ( Ru 4 ( Cu Cu 9 ( L L 5. T tructur of t quaton (0) alow to u an ffcnt traton algortm (Nwton Rapon algortm Katznlon algortm [ 3]). W rmar tat from an traton to t otr mut b cangd only t paramtr aocatd to t nonlnar crcut lmnt. T Jacoban matrx n Eq. (0) can b obtand mpl from t lop of t approprat gmnt of t nonlnar cur. (9) 4. CONCLUSION T ybrd analy mtod of t nonlnar analog crcut prnt t adantag tat t allow t computaton only onc at t bgnnng of t traton

9 9 Gnrald ybrd mtod for nonlnar analog crcut analy 40 proc of to part of t crcut quaton tat xcluly dpnd on t paramtr of t lnar lmnt. Combnng t ybrd procdur wt a ry ffcnt mplct ntgraton algortm n wc only t ymbol of t paramtr corrpondng to t nonlnar crcut lmnt ar condrd a gnfcant ffcncy n crcut analy and an mpromnt of t accuracy n t numrcal calculaton ar obtand. T procdur of t ybrd matrx gnraton ry uful for tady-tat rpon computaton and t may b uccfully ntgratd n t frquncydoman approac bad on armonc balanc and lat quar mtod. T mtod rmarabl by t grat ffcncy and gnralty. ACKNOWLEDGMENT T rarc fnancd from CEEX 258 grant. Rcd on 7 Fbruary 2008 REFERENCES. L. O. Cua P. M. Ln Computr-Add Analy of Elctronc Crcut: Algortm and Computatonal Tcnqu Englwood clff N J: Prntc-Hall B. J. Lon D. J. Safr Voltra r and Pcard traton for nonlnar crcut and ytm IEEE Tran. on Crcut and Sytm CAS-25 Spt. 978 pp J. M. Ortga W. C. Ronboldt Itrat oluton of nonlnar quaton n ral arabl Nw Yor Acadmc Pr A. Uda L. O. Cua Frquncy-doman analy of nonlnar crcut drn by mult-ton gnal IEEE Tran. on Crcut and Sytm CAS-3 Spt. 984 pp M. Iordac S. Cra A nw dtrmnaton mtod of t ordr of complxty for a nonrcprocal lctrc crcut R. Roum. Sc. Tcn. Élctrotcn. t Énrg. 8 pp (983). 6. M. Iordac D. Dorncu On t ybrd mtod for nonlnar rt ntwor analy R. Roum. Sc. Tcn. Élctrotcn. t Énrg. 4 pp (982). 7. M. Iordac Luca Dumtru I. Mat SYMNA Symbolc Modfd Nodal Analy Ung Gud Elctrcal Dpartmnt Lbrary Bucart P. Wambacq G. Gln J. Grrt Low Powr Dgn Tcnqu and CAD Tool for Analog and RF Intgratd Crcut Kluwr Acadmc Publr Boton Dordrct London M. Iordac Luca Dumtru Computr Formulaton of Symbolc Stat Equaton for Analog Nonlnar Crcut wt Exc Elmnt Procdng of t Intrnatonal Sympoum on Nonlnar Tory and t Applcaton NOLTA pp M. Iordac Luca Dumtru Effcnt Dcompoton Tcnqu for Symbolc Analy of Larg Scal Analog Crcut by Stat Varabl Mtod Analog Crcut and Sgnal Procng 40 3 pp M. Iordac Luca Dumtru Crcut Daoptc Analy Bad on Ordr Rducton of t Stat Equaton Procdng of t Intrnatonal Sympoum on Sgnal Crcut and Sytm ISSCS pp