4. Voltage Induction in Three-Phase Machines

Transcription

1 4. Volage Indion in Three-Phase Mahines NIVERSITÄT 4/1

2 FARADAY s law of indion a b Eah hange of flx, whih is linked o ondor loop C, ases an inded volage i in ha loop; he inded volage is he negaive rae of hange of he linked flx. If oil is sed insead of loop wih N series onneed rns, so i is N-imes bigger: i N Flx linkage d / d i d / d Ψ N Φ d / d Changing of : a B is hanging, b Area A is hanging wih veloiy v i Flß: Φ BdA A NIVERSITÄT 4/

3 Indion in resing and moving oils Resing oils Flx densiy B is hanging wih ime i Coil a res i Eleri field srengh E Wi Transformer oils Saor oils of AC mahines d d Transformer indion ( E Eleri field srengh Wi B/ Appliaion of FARADAY s law: Moving oils Flx densiy B is onsan wih ime Coil moving wih veloiy v d / d N d / d / EWi ds i v B ds E d d i A B da NC E Wi Aons. E b E b v B Roaing armare of DC mahines b Roaing indion ds B da ( v B ds (Derivaive of prod! C B d ds N da N v B ds ( d A C NIVERSITÄT 4/

4 Example: Inded volage in simple linear mahine Coil (nmber of rns N, oil span moves wihin air gap beween iron yoke and permanen magnes (Poles N-S-N-S, Pole widh b p = wih veloiy v. b i derived from hange of oal flx linkage: observer ress wih oil: i = -d/d : (ALTERNATIVE CALCLATION TO a! Flx linkage hanges d/d, bease oil moves, giving hange of oil o-ordinae x = v! Coil flx linkage: Inded volage: Fai : Inded volage i may be ALWAYS derived from hange of oal flx linkage. NIVERSITÄT 4/4 i N perpendilar l o eah oher: (v B ds N vbl N B da N l A i 0 xb xb N lb ( x d / d N lb d( v / d N a i inded in moving oil: B / : no hange of flx densiy. Loop C only onsidered along lengh l, as winding overhang oside of magnei field. v, B, s 0 vb l

5 Law of indion: also alled: "LENZ s rle" Lenz s rle: A hange of flx linkage indes volage i, whih drives a rren i in he loop, whih exies a magnei field B e, whose direion is opposie o he original hange of flx linkage. Example: Indion in shor iried loop a res: - The hange of exernal field B ases an inrease of flx densiy wih orienaion from boom o op. This ases inrease of flx in loop area A and indes elerial field E Wi. B / - E Wi is lef hand oriened o and drives in loop C a rren i. - Crren i exies (Ampere s law! a righ hand oriened magnei field B e. B / - Orienaion of B e is opposie o hange of original flx densiy. Fai: The reaion field B e as AGAINST he original flx densiy hange! NIVERSITÄT 4/5

6 Indion of volage in saor oil Sinsoidal moving wave τ / Φ( l p B (x,dx B x, Bˆ os( x / p 1( 1 τ lbˆ osω δ1 p δ1 π τ p/ Inded AC volage in oil is sinsoidal: Volage amplide: ˆ i, N fn ases hanging oil flx ( flx linkage Ψ( ( d( / d ˆ i, i, ˆ plb 1 N Φ( sin (fll-pihed oil NIVERSITÄT 4/6

7 Inded volage by fndamenal and harmoni waves Roaing roor field (speed n: is a FORIER-sm of fndamenal and harmoni waves: x B x Bˆ, (, os(, 1,, 5, 7,... n p p p / p ˆ ( l B ( x, dx lb sin( os( AC oil flx: Inded volage: Fai: NIVERSITÄT p /, d p lbˆ 4/7 sin( sin( i,, N N d In saor oil no only sefl volage de o fndamenal (freqeny f = n p is inded, b also harmoni AC volages wih smaller amplides, b inreased freqenies. Smaller volage amplides proporional Noe: sin( / ( 1 ( 1/ hanges only sign, b no amplide., harmoni freqenies f = /(. Bˆ wih = 1,, 5,... gives only 1, -1, 1, -1,.... Expression

8 Example: No-load volage in fll-pihed oils 1-pole synhronos generaor: n = 500/min, p = 1, fll-pihed oils, saor oil daa: N =, W = p = 0.5 m, l = 1 m Fndamenal freqeny of inded volage: f n p ( 500/ Hz Inded harmoni volage amplides depend on roor air gap field amplides : Bˆ Bˆ i, ˆ i, Bˆ / Bˆ f 1 / i, / i, 1 - T % Hz mwb V % Fai: Amplide spera of inding field and inded volage are idenial: For a fllpihed oil he spaial field disribion and he ime fnion of volage are idenial! NIVERSITÄT 4/8

9 Indion of volage in pihed oil Pihed oil: Coil span is only W insead of p : Φ ( l W / W / Bˆ x os( p dx p lbˆ W sin( p os Linked oil flx is smaller by pih oeffiien k p,, ompared o fll-pihed oil. k p, W sin p NIVERSITÄT 4/9

10 Indion of volage in grop of oils = 1: The inded sinsoidal AC volage per oil grop is he sm of omplex phasors of he q oils. The oil volage phasors are phase shifed by angle beween adjaen oils: Disribion oeffiien: k d, ˆ i, qˆ gr, i,, Q, sin q Q, q sin Q, sin m q sin mq Q, /(mq NIVERSITÄT 4/10

11 Inded volage per phase Mahine wih p poles, wo-layer winding: One phase onsiss of p oil grops wih q pihed oils per grop. Inded volage per phase (r.m.s. vale: Fndamenal: ˆ i1 f N kw1 plb 1 N pqn, / a p -h harmoni: f N k lbˆ i, w, Example: 1-pole synhronos generaor: n = 500/min, p = 1, f =50 Hz - Saor winding: N =, q =, W = 5/6 p, a = 1, p = 0.5 m, l = 1 m - Nmber of rns per phase: N pqn / a 1/1 48 k w1 d1 p1 Bˆ Bˆ / Bˆ 1 f i, i - T % Hz mwb V % k, / i, 1 Fai: By pihing and by oil grop arrangemen volage harmonis are reded drasially. k NIVERSITÄT 4/11

12 NIVERSITÄT 4/1 Sar onneion: no hird volage harmoni 0 ( ( ( ( ( V V If he saor winding is sar onneed, he hird harmoni volages in all hree phases, V, W are IN phase and IDENTICAL! Therefore he line-o-line volages do no show rd harmoni volage omponen. Phase volages in phase ase IN PHASE rd harmoni rrens, whih CANNOT flow a isolaed sar poin (de o nd Kirhhoff s law 0 0 / I I I I I Z I W V V W V ( os( ˆ / 4 os(( ˆ ( ( os( ˆ / os(( ˆ ( os( ˆ ( W V

13 Sar onneion: no hird volage harmoni Measred no-load volage line-o-line of a 4 pole PM synhronos generaor a 1000/min, q =, skewed slos, sar onneion, showing nearly ideal sine wave bak EMF Forier-Analysis of no-load volage: = 1:.5 Hz, 74.8 V = 5: 167 Hz, 0.4 V Oher amplides > 5 are negligible! NIVERSITÄT 4/1

14 Three phase winding: Self indion leads o magneizing indane Saor air gap field waves, exied by saor rren I, inde in saor winding by self indion he volage i! B x x Bˆ m k (, os B N I ˆ 0 w, 1, 5, 7, 11,1, p p Saor air gap field waves B ( x, : Speed n is n syn /. Hene saor field fndamenal and field harmonis inde in saor oils ALL wih he same freqeny f. f p( n / syn pn r.m.s. of self-inded volage per phase for eah -h field harmoni: i, f N kw, syn p lbˆ Magneizing indane per phase: L h for-h air gap field harmoni wave. f L I i, h L kw h N, 0 m l p p NIVERSITÄT 4/14

15 Sray indane of saor winding per phase L L L Air gap field: Fndamenal wave = Magneizing field (sbsrip h: Magneizing indane L h Magnei field in slos (slo sray field and arond he winding overhang is NOT linked wih roor winding. I does NOT prode any fores wih roor rren. Hene i does NOT onribe o eleromehanial energy onversion, and is hs alled sray field (sbsrip. NIVERSITÄT 4/15 L h L h Sray flx indes in saor winding addiional volage by self indion. Hene we define: Slo sray indane L Q, overhang sray indane L b : ( L L, 1 i, Qb Q b Air gap field harmoni waves inde saor winding wih volage i, wih he same freqeny f. So hey are smmarized as oal harmoni volage : L, Q, b, o i, 1, 5,7,... i, 1, 5,7,..., Lh, oal Lh ( 1 o Lh, 1 I kw o 1 1, 5,7,... 1, 5,7,.. kw, 1 o : harmoni sray oeffiien (is small: a Harmoni field waves are linked o roor, b disrbe basi mahine fnion; hene hey are smmarized in harmoni sray indane L o : L I L L i, o, o,, o o h I

16 Field lines B of a age indion mahine Saor Main flx NORTH POLE Main flx: Links saor and roor winding; field lines ross he air gap Leakage flx (sray flx: Is only linked wih eiher saor or roor winding; field lines DO NOT ross he air gap Roor Air gap Roor slo leakage flx Example: For-pole wedge bar roor: Field lines a sand sill (n = 0 - Roor freqeny = Saor freqeny - Roor rren is NEARLY in phase opposiion o saor rren Saor slo leakage flx NIVERSITÄT 4/16

17 Three phase winding in saor and roor In saor and in roor eah a hree-phase winding is arranged: - in saor: phases beween erminals -X, V-Y, W-Z, sbsrip s, - in roor: phases beween erminals -x, v-y, w-z, sbsrip r. We assme: Roor is a res (sand sill, and is rned by angle wih respe o saor. = angle beween winding axis of saor and roor winding (= enre of oils. NOTE: =, if roor is shifed o saor by poles: p. Pole nmbers of saor and roor winding ms be idenial p! NIVERSITÄT 4/17

18 Mal indane beween saor and roor phase Saor Roor From now on only fndamenal Pole on p p field waves onsidered! Phase on m s m r Trns/Phase N s N r Pihing W s / p W r / p Coils/grop q s q r Slo on Q s Q r Qr Q s Mal indane: e. g.: Saor air gap wave B ( x, indes volage in roor winding: (, ˆ x ms B x B os( s wih amplides Bˆ 0 NskwsIs p p Amplides of inded volages in roor winding: f N k i, r s r wr plb ˆ Roor freqeny f r (a loked roor = sand sill: f r = f s. Fndamenal wave: Mal indane per phase M sr : M i, r s sr I s M sr m l s 1 p 0Nskw, snrkw, r Noe: sr M rs p M a ms mr! NIVERSITÄT 4/18

19 Roary ransformer Inded roor volages are phase shifed by angle wih respe o saor volages, as roor is shifed by ha angle mehanially. Series onneion of saor and roor winding and (in he same way: V and v; W and w The following resling volage ors beween FIRST erminal of saor winding and SECOND erminal of roor winding (per phase: j j j s r s re, e. g. r = s : s se s 1 e By rning he roor we ge a oninos hange of angle. Fai: Wih roary ransformer a oninos hange of op volage beween 0 and s is possible a onsan line freqeny, whih is sed in es failiies as variable volage sore. NIVERSITÄT 4/19