Parks Equations Generalised Machines. Represent ac machines in the simplest possible way.

Transcription

1 Park Euaton Generale Machne ereent ac machne n the mlet oble way. All machne excet oubly alent reluctance machne e.g. SM an teng motor are eentally alternatng n nature Ueful torue from funamental (atal) comonent of electrc an magnetc fel. Sngle ace harmonc = two uantte magntue an angle or n hae an uarature. Magnetc fel from multhae wnng rotate wth contant magntue Smlet way to rereent from reference ont ynchronou wth fel.

2 e.g. Inucton motor tator teay tate - hae, alternatng current, alternatng flux rotatng fel Smlet rereentaton = col wth c current rotatng wth fel Th exactly what a haor agram oe magntue an angle, ynchronouly rotatng

3 Synchronou machne

4 C Machne

5 Tranform e.g. Inuctor wth N turn. Want to rereent that wth an euvalent col wth turn. We have v ol ol ol t. If we go from N turn to we ntnctvely know that we mut ecreae the volt by N an ncreae the current by N. To kee the euaton rght we mut ajut the meance by N.N v v ol new new new N v N N ol t N new new Formally - tranformaton for ay current..e. ol C new Nee ytem to be ame after tranformaton.e. kee comlex ower contant - o calle ower nvarance - ctate voltage tranformaton.

6 Conton for Power Invarance t t v v ol t* v C t S P jq v S S Power In var ance ol ol ol new * * new ol new new ol t* new ol [ S] C v *

7 Orgnal voltage euaton wa: v Z ol ol ol t* new ol ol [ S] C Z t* C Z C new ol new t* t* C Z C new t* v new new ol new t* Therefore v new C Zol C new v new Z new new t* Z new C Zol C t* vnew C vol t* v C v ol new

8 Orthogonal Tranformaton If C t* C then v C ol v new C ol new C new ol

9 : Phae Tranformaton The, col nee to mmc the mmf from the a,b,c col aume ame turn n, col a n a,b,c an frt harmonc only:- c b c b a c b a c b a co(5) co() co(9) co(4) co() co() C ol new a b c

10 Prefer other way roun o nvert a b c Nee to a zero euence = comonent of current whch ame n all three hae. [All comonent ne wave frt harmonc n tme] Three hae of zero euence a u to zero n ar ga {but trlen ace harmonc on t cancel.e. r, 9 th,5 th etc. all ele o} Zero euence only ext f:- Searate hae connecton. Star connecte wth a neutral correcton, neutral current = x zero euence current.

11 C zero euence C a b c o o o o a b c V V V V V V a b c o a b c Voltage an current tranformaton ame excet for /. Make ame - remove / from voltage tranformaton. Gve error n ower an torue - to correct - multly by /. Naty but t work.

12 Force tranformaton to be ame an have ower nvarance by changng the turn by.e., col have le turn than the a,b,c col an zero euence turn by :- V V V V V V o a b c o a b c Th an orthogonal tranformaton.e. C C t *

13 otatng to Fxe Axe Tranformaton e.g. Synchronou or Inucton machne nee frt ::- then rotatng to fxe:- co n n co v v co n n v co v Note that [C] t* =[C] - for th tranformaton

14 Generalze Machne C Machne Ue tanar form:- Q = tatonary col = ueo tatonary.e. commutator col v v P a P v t v a t =Fel, =armature, f a f a t

15 ax v v v Q Q v ax Take a four col C machne v Z. Q a a a a a Q Q a Q v v v v

16 Alcablty 4 col machne ue to rereent we range of machne. e.g. Armature col n ynchronou machne Tranform them from col to, then tranform to col whch are tatonary wth flux Th exactly what the commutator oe hycally t tranform the real rotatng col n the armature whch rotate through the flux to an euvalent col whch tatonary wth reect to the tator. Inucton motor - armature col to, n rotor bar to col, fxe to tator, rotor or ynchronou

17 Park Euaton Wrte n form ung flux lnkage - Park euaton We note (for our 4 col generale machne) that a Q a v v therefore Frt ale to ynchronou machne n9. Blonell n France an Park, Concora, Kngley an Kron n Amerca

18 Torue Euaton We can wrte our voltage euaton a: v t a a a G Q a G t t t t G t T G r Ohmc ate of change of Store Energy a a Power Outut Through Shaft o P v t But r P t where P =ole ar T P G

19 Store energy Statonary axe - nuctance contant wth tme.e. t ate of change t t t t

20 T P Q a a Q a Q T P a Q T P a a Q P

21 Synchronou Machne ereente ung Q Q axe go wth the rotor, reuce tator to col whch are ueo tatonary wth reect to the rotor.e. the tator col are attache to a commutator wth the commutator bruhe attache to the rotor v Stator col rotataton ax v v

22 a a a v v v In the teay tate.e. t V I I V I I X V I I I V I E I X a f

23 Full Tranent Moel For a Synchronou Machne To to ocllaton (calle huntng) aroun Hz a hort crcut wnng - amer or amorteur wnng try to to flux movng w.r.t. rotor..e. amen or kll rotor flux ocllaton. Wnng ame a urrel cage wnng n nucton motor - try to to flux lng. Stator col rotaton,

24 [ Z] a a k a k kk a k k kk a a a a a a a a a efer all wnng to ay the, col by turn rato uare then a l l f fel leakage nuctance., a l l armature leakage nuctance a l, kk a k k ax amer leakage nuctance. kk a k, k ax amer leakage nuctance.

25 Synchronou Machne / Euvalent Crcut a X l X f f f X a k v v f Xk ax

26 I a X l X a k V Xk ax

27 Inucton Motor Moel Stator eference Frame otor - t N rotor bar to two hae rotatng at r (electrcal) n otatng to fxe Stator - :

28 m m ( ) ( ) ( ) m m ( ) r r r m m r r r rotor r, tator,, Q r a m a Synch. fre., r, r rot. elec. ee. Steay tate = j

29 otor eference Frame Stator :, rotatng fxe, otor N: m m m m m r r m r r Ue for Inucton machne wth l rng connecton (e.g. FIG oubly fe nucton generator) - euaton n actual rotor uantte at l freuency. Steay tate = j

30 Synchronou eference Frame Stator :, rot. to fxe, otor n:, rot. to fxe, r r r m m r r r m m m m m m

31 In teay tate all fluxe rotate at ynchronou ee - n ynchronou reference frame all AC uantte become C. = gvng r r m r r m m m