Multi-Machine Systems with Constant Impedance Loads

Transcription

1 Mult-Mach Systms wth Costat Impac Loas Th parts of th txt whch w hav yt to covr clu: Chaptr 3: Systm rspos to small sturbacs Chaptr 6: Lar mols of sychroous machs Chaptrs 7-8: Exctato systms a Effct of xctato o stablty Chaptr 9: Multmach systms wth costat mpac loas Chaptrs -3: Sp govrg a prm movrs (stam/hyro/cts/cc uts) W wll stuy th frst part of chaptr 7 (7.-7.3) a o part of chaptr 8 (8.3) o xctato; w wll ot stuy ay of chaptrs - 3 (turb-govrors) at all. W wll sp som tm o Chaptr 9 a th mov back to Chaptr 3. W wll vry brfly look at Chaptr 6 as wll (ot that Chaptr 6 s to Chaptr 3 as Chaptr 4 s to Chaptr,.., Chaptr 4 xts th covrag of trast stablty aalyss o Chaptr from th classcal mach mol to mor laborat mach mols. Chaptr 6 os th sam thg, xcpt sta of trast stablty, t xts th covrag of small-sgal stablty o Chaptr 3). So hr w look at Chaptr 9. Chaptr 9 cossts of th followg sctos: 9.: Itroucto 9.: Problm statmt 9.3: Matrx rprstato of a passv twork o Ntwork th trast stat o Covrtg to a commo rfrc fram 9.4: Covrtg mach coorats to systm rfrc 9.5: Rlato btw mach currts a voltags 9.6: Systm orr 9.7: Machs rprst by classcal mthos 9.8: Larz mol for th twork

2 9.9: Hybr formulato 9.: Ntwork quatos wth flux lkag mol 9.: Total systm quatos 9.: Multmach stuy W wll work sctos Not that Payar s book also gvs tratmt of ths pp Scto 9., Itroucto: I wll us ths scto to mphasz th mportac of loa molg. Plas ra th 993 Task Forc papr o loa molg post to th cours wbst. Also, plas rvw th WECC ocumt o compost loa mol spcfcatos, also post to th wbst. Ths lattr ocumt shows th wll-kow llustrato us for compost loa molg, show blow. A mor rct llustrato llustrats that t accommoats strbut PV, as show blow. Larg 3- Motor Aggr. Mum 3- Motor Aggr. Small 3- Motor Aggr. All - Motors Aggr. Expotal Loa Aggr. Loa Shg Schms ZIP Loa Aggr.

3 Thr ar two basc typs of commoly us loa mols. Statc: o Expotal o Polyomal Iucto motor Th polyomal s probably th most commo. O vrso of th polyomal s th so-call ZIP mol: V P P A B V V P P D E V V C V V F V LPf Typcally, th frqucy sstvty coffcts oby <LP<3 a <LQ< so that wh frqucy cls (mag f<), P crass a Q crass, whch ts to b th cas for a ucto motor. Th voltag sstvty coffcts must oby A+B+C= a D+E+F=. If w st A=B=D=E= a C=F=, th w hav a costat mpac mol. Ths loa mol provs that powr cosumpto of loas crass as voltag rops. Ths charactrstc typcally crass th svrty of systm rspos trms of trast stablty that: W usually s voltag rop urg a aftr a sturbac Wh voltag rops, costat Z loas cosum lss powr accorg to th squar of th voltag rop whch tur mprovs th stablty prformac of th grators. O avatag to usg th costat Z-mol s that t allows us to asly ruc th twork to grator os as all loas ar rprst th -bus. W obta th mpac quvalts va Z= V /S *. LQf 3

4 O shoul ot carfully hr th ffrc btw loa molg for trast aalyss a loa molg for stay-stat aalyss. Typcally, for stay-stat aalyss (usg powr flow), w rprst th loa usg costat powr mols. Som powr flow programs o allow for usg othr loa mols,.g., ZIP. Howvr, f your powr systm cotas ur-loa-tap-chagg (ULTC) trasformrs coctg btw th trasmsso systm a th loa (most commoly btw th subtrasmsso a th strbuto systms), a most o, th us of aythg xcpt a costat powr mol s usually approprat ulss you ar also rprstg th ULTC trasformrs. Th raso for ths s as follows: Stay-stat aalyss of sturbacs usg powr flow s typcally o to aalyz th 3- mut tm pro followg th sturbac. Th valu of 3 muts s chos bcaus ths s ough tm for th ULTC to oprat fully, rstorg th voltag lvls th strbuto systm, so that th loas actually s a costat voltag a thrfor bhav as costat powr loas. Scto 9., Problm statmt: Each mach s rprst by th followg rlato: x f ( x, v, Tm, t) (9.) whr x s th stat vctor (coul b ay umbr of stats btw -8 pg o th choc of mach mol), v=[v, vq, vf] T, Tm s th mchacal torqu, a t s tm. Rcall that th put vctor for ach of our mach mols clu v a vq (or V a Vq whr v V a 3 Vq ), whch ar th - a q- axs compots of th mach trmal voltag. For xampl, th currt-stat-spac mol for mol s: v 3 4

5 ) ( m G Q q D F G Q q q D q F q G Q q D F T v L D L L N R L whr ; D F G Q q G Q D F L L N r r r r r r R G G Q Q G Q q D R D R F F D F L M M L L L M M L L L G Q q D F q F v v v v ; Ths trms v a vq (or V a Vq) ar trm by th twork, a w thrfor to trfac th mach mol wth th twork orr to accout for thm. W assum hr that vf a Tm ar fx (thy ar actually govr by th xctato cotrol a th turb-govror cotrol; w wll stuy cotrol of vf ths cours, Chaptrs 7-8, but w wll ot hav tm to stuy cotrol of Tm, Chaptrs -3). Lt s assum that w ar usg th currt stat-spac mol of Mol (whch s th full mol clug th G-crcut a two ampr wgs, so t s call mol.). Not that A&F mak th followg statmts at th bgg of Scto 9..

6 Cosr th st of quatos (9.). I th currt mol vlop Chaptr 4, t rprsts a st of sv frst-orr ffrtal quatos for ach mach. Our currt mol actually ha ght frst-orr ffrtal quatos for ach mach, sc w clu th G-crcut. Th umbr of th varabls, howvr, s : fv currts, a, a th voltags a. q A so, wth a 8-stat mol (wth G-crcut), w hav th umbr of stat varabls s ght: sx currts, ω a δ; a th umbr of varabls s t: ght stat varabls a th voltags v a vq. Assumg that thr ar sychroous machs th systm, A assumg that all machs ar mol wth th 8-stat mol w hav a st of 7 ffrtal quatos wth 9 ukows. Wth th G-crcut, w hav a st of 8 ffrtal quatos wth ukows. Thrfor, atoal quatos ar to complt th scrpto of th systm. That s, th varabls v a vq rsult th atoal two ukows pr mach, a so w a atoal two quatos pr mach. Ths quatos ar obta from th loa costrats. Our obctv s to rv xprssos for v a vq trms of th stat varabls (a so avo ag atoal varabls), whch th cas of th currt stat-spac mol of Mol (wth G- crcut), woul b th sx currts, ω a δ. W wll o ths from th loa costrats. W bg by rcallg th stator-s quvalts to v, vq,, a q, gv by: 6

7 V v 3 Iq 3 whr subscrpt cats that th rlatos apply to mach. I 3 V q v q 3 W also hav that V V q V for vry mach =,,. I I q I (9.) Thus w hav a vctor of oal voltags a currts for vry grator bus gv by: V V V q q V V V V I I I q q I I I I (9.4) (Not that w us urls to ot vctors a matrcs, a w us ovrbars to ot phasors). Our problm s to xprss V trms of I. O mght thk that ths s a asy problm, bas o rcollcto of th -bus rlato whch has that I=V. Howvr, thr s a maor ssu og ths 7

8 Th lmts of ths two vctors,.g., Vq+V a Iq+I, ar, by fto, xprss o th -q rfrc fram of th corrspog machs. W hav o othg at ths pot to rlat th -q fram of o mach to that of aothr. A&F say t ths way (p. 369, talcs a): Not carfully that th voltag a th currt ar rfrr to th q a axs of mach. I othr wors th ffrt voltags a currts ar xprss trms of ffrt rfrc frams. Th sr rlato s that whch rlats th vctors V a. Wh obta, t wll rprst a st of complx algbrac quatos, or ral quatos. Ths ar th atoal quatos to complt th mathmatcal scrpto of th systm. V So th lmts of V (a th lmts of I) ar xprss o ffrt rfrc frams. Ay aalyss usg ths umbrs as s woul hav rlatv agls btw os th twork that ma absolutly othg. Sc rlatv agls hav a vry larg ffct trmg powr flow, ths uaccptabl. I I 8

9 Scto 9.3, Matrx rprstato of a passv twork: I cosrato of a multmach systm Chaptr, usg th classcal mach rprstato, bcaus th mach tral EMF s costat, w coul ruc th twork to ts tral mach os, thus lmatg th os corrspog to ach mach s trmal voltag Va. Now, howvr, w to rta th o corrspog to ach mach s trmal voltag Va bcaus all of our hghr-orr mols rqur t through th prsc th mols of v a vq. Th ffrc btw ths two approachs ar llustrat by th Fg. blow from your txt (lft, tral os, Fg..7, a rght, trmal os, Fg. 9.). Fg. Th w assum, for ow, that w rprst all loas usg costat mpac shuts. Th w us twork ructo (Gaussa lmato) to lmat all twork os xcpt th mach trmal os. W hav alray rcogz that w caot xprss I=V usg q. (9.4) bcaus th varous vctor lmts ar all o ffrt rfrc frams. 9

10 So lt s cosr a w st of oal voltags a currts that ar xprss to a commo rfrc fram whr o of th quatts, oft o of th voltags, has a agl sgat as. W wll rfr to ths st of oal voltags a currts as a, artculat as V-hat a I-hat. So th url cats vctor, a th hat cats that all lmts ar rfrr to th twork rfrc fram. So o th twork rfrc fram, t s accptabl to wrt that Iˆ Vˆ (9.5) whr s th twork amttac matrx. Of cours, at ths pot, w ar smply cocturg that w ca xprss all voltags a currts to a commo rfrc fram, but w hav ot yt o t. But Dr. Arso s carful.. h rcogzs that q. (9.5) s a stay-stat rlato, a h taks a lttl as to chck: ur what cotos ca w us q. (9.5) for trast aalyss? To aswr ths qusto, scto 9.3., h wrts th tm-oma voltag rop quato for a twork brach, a th trasforms ths quato usg Park s trasformato. Ths trasformato s bas o a assum sychroously rotatg rfrc fram whch, at t=, s alg wth th a-phas of a chos mach. Ths acto, th, locats th mach s rotor, a thus th mach s -axs, at R t / Fg. llustrats. Vˆ Î

11 -axs a-phas axs π/ q-axs δ ωrt Sychroously rotatg rfrc Fg. I wll ot go through ths aalyss but rathr wll smply stat th coclusos. Dr. Arso s cocluso s that: Vk ( ) zk I k ( ), k=,, b (9.6) whr a ar th brach voltag rops a brach currts, rspctvly, xprss o th -q axs rfrc fram of mach, that s, th rfrc s th q-axs of th th mach locat at agl δ wth rspct to a sychroously rotatg systm rfrc, zk s th mpac of brach k, a b s th total umbr of brachs th twork. I k ( V k () ) Equato (9.6), whch s our staar Ohm s Law rlato, s applcabl for trast aalyss f th followg two cotos ar satsf:. Th frqucy, a thrfor th ractacs of th brachs, ar costat.. Currt rvatvs ar much lss tha sp-currt proucts.

12 q q Ths s aalogous to whr w assum that trasformr voltags ar much lss tha sp voltag rops (sv),.., th -q voltag compots u to trasformr acto (.., varato -q currts or -q flux lkags) s much lss tha th -q voltag compots u to th sp. W us ths rvg th E mol, xprss as: q I ato to tfyg th cotos ur whch w ca us our famlar stay-stat form of Ohm s Law (a thus th -bus rlato), q. (9.6) also provs that w may xprss th twork to a partcular mach s -q rfrc fram. But ths os ot o us too much goo sc w hav all th mach mols xprss to thr ow fram. So a bttr approach s to xprss all of th mach -q rfrc frams to a twork rfrc fram. Lt s try that (Scto 9.3.). W hav alray f th -q rfrc fram of th mach. Now w f th twork rfrc fram, a w wll ot th twork rfrc fram as D-Q (o NOT cofus ths otato wth th uppr-cas D,Q otato us for th ampr wgs!!!!). So our qusto s: how to covrt a voltag (or currt) o th -q rfrc fram to a voltag (or currt) o th D-Q (twork) rfrc fram? q

13 Fg. 3 (Fg 9.4 txt) llustrats. D Vqsδ V, ˆ V q V Vsδ V Vcosδ V q Vqcosδ Q Fg. 3 Not two thgs wth rspct to Fg. 3: V, ar raw lag th q-axs, whras w kow that for grator acto, th trmal voltag wll lag th q-axs. Ths s bcaus Fg. 3 s raw to facltat urstag of how to proct ay gral quatty gv o th -q fram to a quatty gv o th D-Q fram. It s ot raw to pct th oprato of a grator. Th agl δ has a w fto. o Whras prvously w hav f δ as th agl by whch th mach tral voltag (a thus th q-axs) las th (sychroously rotatg) mach trmal voltag; o ow, Fg. 3, w f δ as th agl by whch th mach tral voltag (a thus th q-axs) las th (sychroously rotatg) Q-axs twork rfrc fram. Vˆ 3

14 From ths pctur, t s asy to s how to comput VQ a VD from Vq a V. It s mportat to rcogz that w ar NOT gttg VQ a VD from (or ar th -q axs compots of (or Vˆ ). V Vˆ ) rctly but rathr gttg t from V from Vq, whch For xampl, cosr gttg VQ. By spcto, w s that V V cos V s V Whr, aga, w mphasz that th agl s th agl by whch mach q-axs las th sychroously rotatg twork rfrc fram. Smlarly, cosr gttg VD. Aga, by spcto, w s that: Thrfor, th voltag fram, bcoms Vˆ V V Q D Vˆ V D wh xprss to th twork rfrc, xprss as: ( V whch ca b factor to prov: Vˆ V V ( V V )(cos s ) V Q I summary, th trasformato that w ar makg s from o st of coorat axs (whr th postv q-axs s assg grs), to aothr st of coorat axs Vˆ V D Q q V V whr th postv Q-axs s assg grs. Hr, th +q-axs las th +Q axs by grs. q q cos V q s V Q s ) V D cos ( V q s V cos ) 4

15 A w hav fou that Vˆ V (9.7) As a xampl, cosr a arbtrary quatty V 3 (xprss o th -q fram), a lt q la Q by =. Th ˆ V 3 V 5 whch s llustrat Fg. 4 blow. V V ˆ q +Q Fg. 3 Bfor w go furthr, lt s clarfy two thgs:. What s th agl δ?. How o w tfy th systm rfrc? W wll tak ths qustos o at a tm.. What s th agl δ? Svral commts hr: a. Valu vs. varabl: I ots o Smulato of Sychroous Machs, w locat th tal valu of δ (for ach mach ) by fg E a. But mak sur you ar clar your m that ths valu (w coul call t δ) s a tal coto, a as such, w ca rfr to t as a spcfc valu; gral, δ s a varabl ( a stat varabl); hr, Chaptr 9, w o logr thk oly of δ as a tal coto but also (a prmarly) as a varabl that wll vary through th cours of our tm-oma smulato. 5

16 b. Th mag of th agl δ has b chag. To ursta ths, w wll rvw what δ was (pot c blow) a what δ s ow (pot blow). c. What δ was: It s worth gog back to th bgg of chaptr 4 to mak sur w ursta what δ was. O p. 84, w wr show th blow agram. F E V It s usful to rvw what A&F sa about ths fgur (pp ), whch I hav cop out blow, quots, wth (my) atoal commts hghlght yllow. Th ma fl-wg flux s alog th rcto o th -axs of th rotor. Ths s F, whch I a to Fgur 4.. It proucs a EMF that lags ths flux by 9. Thrfor th mach EMF E s prmarly alog th rotor q-axs. I also a ths to Fgur 4.. Cosr a mach havg a costat trmal voltag V. For th grator acto th phasor lag th phasor V. I also a ths to Fgur 4.. E shoul b 6

17 Ky pot: Prvously, th agl δ has b th agl by whch E V. las Th agl btw E a V s th mach torqu agl δ f th phasor V s th rcto of th rfrc phas (phas a). At t= th phasor V s locat at th axs of phas a,.., at th rfrc axs Fgur 4.. Th q- axs s locat at a agl δ, a th -axs s locat at θ=δ+π/. I hav rraw Fg. 4., for t=, as Fg. 4 blow. +q-axs Phas a axs. rotato E a δ V θ +-axs Fg. 4 At t>, th rfrc axs s locat at a agl ωrt wth rspct to th axs of phas a. Th -axs of th rotor s thrfor locat at θ= ωrt+δ+π/ (4.6) whr ωr s th rat (sychroous) agular frqucy ra/s a δ s th sychroous torqu agl lctrcal raas. I hav rraw Fg. 4., for t>, as Fg. 5 blow. 7

18 Phas a axs. +q-axs δ ωrt θ Fg. 5. What δ s ow: I vlopg a systm sychroously rotatg rfrc fram, δ (for mach ) chags from th agl by whch th mach q-axs las th trmal voltag to th agl by whch th mach q-axs las th sychroously rotatg systm rfrc.. How o w tfy th systm rfrc? To aswr ths qusto, w brfly ump aha to th bgg of Scto 9.4, whr t ras: Cosr a voltag vabc at o. W ca apply Park s trasformato to ths voltag to obta vq. From (9.) V V q V +-axs (9.) ths voltag ca b xprss phasor otato as V, usg th rotor of mach as rfrc. Ths statmt s a lttl mslag. Sc th -axs s alg wth th rotor, usg th rotor as rfrc mpls usg th -axs as rfrc. Howvr, th phasors of (9.) ar xprss wth th q-compots I I q I 8

19 alog th ral axs ( ) a th -compots alog th magary axs (9 ), mplyg th q-axs s th rfrc. It may b that wh A&F wrot usg th rotor as rfrc, thy mat usg th rotor fram as rfrc, whch coul b trprt as usg th q-axs as th rfrc. W wll assum thr that thy mat to cat thy wll us th q-axs as rfrc. It ca also b xprss to th systm rfrc as usg th trasformato (9.7). (9.7) I hav rraw th fgur, as blow, to llustrat: +D-axs Vˆ V --axs +Q-axs V +q-axs δ,w V V q δ,ol +-axs Exprsso (9.7) ca b urstoo as follows Obsrv that th agl of phasor V, tf as δ,ol, a gv o th -q fram, must b gatv (th q-axs las V, a so f w xprss V rlatv to th q- axs, wth th q-axs havg a agl, th agl of V must b gatv). 9

20 O th othr ha, f w xprss rlatv to th Q- axs (to obta ), w obsrv that th agl must b postv as V s lag th Q-axs. W obta ths va: Vˆ V, ol, w (*) Now, ramg δ,ol a δ,w to b cosstt wth A&F, w ca wrt (*) as: V V (**) whch s obta from Vˆ V (9.7) Now rcall th quato rlatg brach voltag rops to brach currts: V z I, k=,, b (9.6) k ( ) k k ( ) Rmmbr what th otato cats that th quatty s xprss to th -q coorat axs of mach. But w wat all quatts o th D-Q (twork) coorat axs, a ow w kow how to achv ths. Vˆ V V Vˆ k Iˆ k I k ( ) k ( ) Substtuto to (9.6) yls: Vˆ k zk Iˆ k A w s that th xpotals cacl so that: Vˆ k zk Iˆ k k=,, b (9.8) Combg (9.8) wth (9.6) w s that Vk V k zk ˆ I ˆ I k=,, b k k I k ( ) k ( ) k k Iˆ V

21 Ths s xpct t says that th rato of a voltag rop across a lmt to th currt through th lmt wll rma th sam f w rotat all voltags a all currts by a partcular agl. Wrtg th abov quato for vry brach th twork rsults th followg matrx rlato: Vˆ z Iˆ V ˆ z I ˆ Vˆ b zbb Iˆ b W may wrt th abov rlato mor compact form: Vˆ b z b Iˆ b (9.9) Som commts about th abov: Sc all off-agoal lmts ar zro, w hav assum that thr s o mutual couplg th twork. (Mutual couplg ca xst, howvr, btw ls that ar physcally paralll a locat clos proxmty, a coto that s fou wh svral crcuts shar a commo rght-of-way.) Th matrx zb s squar wth o-zro valus alog th agoal a s thrfor vrtbl. W ot ts vrs as yb, such that: Iˆ b y Vˆ b (9.) b Th matrx of mpacs xb s call th prmtv mpac matrx, th matrx of amttacs yb th prmtv amttac matrx, a th quatos usg th z- a y- forms ar call th prmtv twork quato, am by Gabrl Kro. Th prmtv twork quato os ot scrb th twork at all,.., t gvs absolutly o formato as to how th vual brachs ar trcoct th twork.

22 I orr to prov twork cocto formato, w th o-cc matrx A, gv by: A a a ab a a a b a a a b whr b s th umbr of brachs th twork. s th umbr of os th twork. ak s gv by: f currt brach k s lavg o a k -f currt brach k s trg o f brach k s ot coct to o Not that A s a b matrx: Numbr of brachs = umbr of rows Numbr of os = umbr of colums P g P g y 4 =- y 3 =- y =- y 3 =- - y 34 = P A g4 P 3=.787pu Lt s ot th oal voltags a currts, xprss to th twork fram, as Vˆ a Î. P =pu

23 Th oal currts may b rlat to th brach currts by summg ovr all currts lavg o. Sc a o corrspos to a colum of th o-cc matrx, w ca rlat th oal currts to th brach currts through a multplcato of A T wth th brach currt vctor,.., Iˆ A T Iˆ b (*) Th matrx A T has ach row corrspog to a o, a thrfor th lmts of ach row wll pck out of th approprat brach flows maatg from that o to prov th total ct currt to that o. Not msos of trms ths rlato, w obta a matrx from th prouct of a b matrx wth a b matrx. So th abov rlato llustrats that th o-cc matrx ca b us to sum quatts. I ths partcular cas, w summ brach currts to gt th oal currts accorg to KCL. What about rlatg oal voltags to brach voltag rops? I ths cas, w cosr KVL a rcall that w to sum th oal voltags to obta th voltag rops. So w to xprss prouct of Vˆ a A som fasho. Î b Vˆ b as a 3

24 If you toy wth ths matrcs from purly a msoal pot of vw, you wll s that AVˆ ˆ V b (**) whr th msos cat that w obta a b from th prouct of a b wth a. W may also rv ths from powr rlatos (rf: P. Arso, Aalyss of Fault Powr Systms, pp ). But w obsrv (**) that ach row of A corrspos to a partcular brach, a th o-zro lmts of that row corrspo to a bus that s coct to that brach. Thr wll oly b two such buss, a th prouct AV wll pck off th two voltags at thr of th brach to f thr ffrc, whch s cota Vb. Substtuto of q. (9.), I ˆ b y Vˆ b,to q. (*), Iˆ b A T Iˆ b,yls: T T Iˆ A Iˆ b A y Vˆ b b (***) a substtuto of q. (**), AVˆ Vˆ b, to (***) yls: Iˆ T T A y Vˆ A y AVˆ b Hr, w clarly s that th famlar -bus (amttac matrx) s obta from th prmtv amttac matrx from: so that w hav, fally, that Iˆ Vˆ (9.) whch rlats oal voltags a currt ctos gv o th D- Q (twork) coorat axs. Now f a squar trasformato matrx T accorg to: b A T y b A b 4

25 T T Th w ca obta th oal currts a voltags xprss o D- Q (twork) coorat axs from th oal currts a voltags xprss o -q (vual mach ) coorat axs from: Iˆ T I Substtuto to q. (9.), T I TV a Iˆ Vˆ Vˆ TV, yls: I T TV MV whr clarly, M T T What os th trasformato o? I MV It allows us to rlat currts th -q coorat fram of o mach, I, I,..., I to voltags th -q coorat fram of all othr machs. ou s, I V os ot work! I MV s th rplacmt w. 5

26 6 Exampl 9.: Th matrx M ca b valuat by prformg th approprat matrx multplcatos: T T M = = ) ( ) ( ) ( ) ( ) ( ) ( whr k= - k. Th gral form of th trm row, col k, th matrx M s: ( ) cos s k k k k k k k k k k k M G B ( ) ( ) cos s cos s G B k B G k k k k k k k k k k F F M G B B G So th -k th trm matrx M s gv by FG+B(k)+ FB-G(k). Ths smplfs for th agoal lmts, sc =, to G+B. So M= FG+B(k)+ FB-G(k) M= G+B

27 Sparatg ral a magary parts, w obta M=H+S whr Hk= FG+B(k) H=G Sk=FB-G(k) S=B ou shoul rvw xampls 9. a 9.3 th txt. Atoal commts: Th ovrall problm s gv by x f ( x, v, T, t) I MV whr M s formulat as follows: A bcaus w hav that M T M T T T m T Now hr s a ssu. If w hav trly costat mpac loas, th all loas ca b clu to th matrx, a th abov formulato s OK. If w hav costat currt loas, th thos loas may b clu th vctor I. A clarly hav both costat mpac a costat currt loas ca b hal by accorg to ths two approachs (us for costat mpac loas a I for costat currt loas). T A y b A A T y b AT 7

28 But f w hav costat powr loas, th thos loas, wh covrt to a costat currt rprstato through I=(S/V)*, ar a fucto of voltag. I that cas, th problm w ar solvg s x f ( x, v, T, t) I( V ) MV whr th algbrac quatos must b solv tratvly. Ethr way, w hav th trfac problm, llustrat a fgur from Bra Stott s papr blow. m Stott, Scto IV of hs papr, troucs a classfcato systm for solvg a ffrtal-algbrac quato (DAE), whch s what w hav. H says that soluto approachs ar charactrz by thr attrbuts:. Th way whch mach a twork quatos ar trfac wth ach othr: a. Partto: altratg b. Smultaous (comb or algbracally). Th tgrato mtho us: a. Explct b. Implct 3. Th tchqu for solvg th algbrac quatos (a ssu f you hav costat powr loas a you solv usg th altratg mtho. 8

29 I prov som cutouts from Stott s papr blow: 9

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