Section 4.2 Analysis of synchronous machines Part II

Transcription

1 Section 4. Anlyi of ynchronou mchine Prt 4.. Sttor flux linkge in non-lient pole ynchronou motor due to rotor The ir-gp field produced by the rotor produce flux linkge with individul phe winding. Thee re determined by the ngulr diplcement between the winding. Conider the following ce in which the grey rrow lwy indicte the direction nd the center of the coinuoidl rotor flux ditribution: Rotor flux long the winding xi The flux linkge f of winding `- due to the rotor field i given by L co co (4..) f f f f Figure 4.. where f i the current in the rotor field winding nd L f i the mutul inductnce between the - nd the rotor winding. f i mximum when 0 nd 80. Note tht L f doe not chnge with ngulr poition for the non-lient pole motor for which the ir-gp length l g i contnt. When the rotor mgnetic field i horizontl, mll movement of the rotor doe not chnge f much nd thu d f dt 0 for 0 nd 80. Rotor field perpendiculr to the winding xi When the rotor pole re verticl ( = 90nd 70), the flux linkge with winding - i Section 4. Syn Motor repreenttion F. Rhmn (EET, UNSW) Nov, 0

2 f f f f Figure 4.. L co co 0 (4..) Note tht the mximum vlue of d f occur for thi ngulr poition of the rotor. dt 4.. Flux linkge of ttor winding due to ttor current Self flux linkge of winding - lekege gp L i L i l L i o (4..3) +i i Figure 4..3 Note tht normlly, L = L bb = L cc Alo, L o = L bbo = L cco Section 4. Syn Motor repreenttion F. Rhmn (EET, UNSW) Nov, 0

3 Mutul flux linkge between winding ` nd `b b L b i b (4..4) bb 0 b b Figure 4..4 Winding b-b i diplced by 0 from the xi of winding -. Hence, frction of the flux produced by winding b-b link with winding -. Note tht the verticl component of bb doe not link with the - winding, only the horizontl component doe. Thu L co 0 i L i b bbo b bbo b L i o b Mutul flux linkge between winding c-c nd - (4..5) Winding c i diplced by 40 from the xi of winding. Hence, frction of the flux produced by winding c link with winding. Thu c L c i c c Lcco co 40 ic L i cco c L i o c (4..6) i c c 0 c cc i c Figure 4..5 Section 4. Syn Motor repreenttion 3 F. Rhmn (EET, UNSW) Nov, 0

4 4.. Repreenttion of non-lient pole (uniform irgp) motor Totl flux linkge of ech phe By neglecting effect of lot, mgnetic turtion, nd uming tht the ir gp i contnt (uniform), the flux linkge of the three ttor winding re: L L i L i L i l o b b c c f L i L L i L i b b bl bbo b bc c bf L i L i L L i c c cb b cl cco c cf (4..7) The elf inductnce of the winding re: L = L o + L l L bb = L bbo + L bl (4..8) L cc = L cco + L cl Self inductnce L, L bb nd L cc re contnt for ll poition of the rotor, becue of the contnt ir gp. The mutul inductnce L b, L bc, nd L c, between winding, b nd c re lo contnt, for the me reon. L b L b L o co 0 Lo (4..9) L nd o c L c L o co 40 L (4..0) becue winding b nd c re diplced by 0 nd 40 degree repectively from winding. The totl flux linkge of winding i thu L L i L i L i o l b b c c f L i L i L i L i o l o b o c f L i L i L i i o l o b c f 3 o l f L L i f Li (4..) where L 3 L o Ll i the ynchronou inductnce of the mchine. t i the inductnce (mgnetic flux linkge per Ampere) of ech phe of the mchine when the fluxe contibuted by ll three phe of the mchine, crrying blnced inuoidl current, re tken into ccount. Section 4. Syn Motor repreenttion 4 F. Rhmn (EET, UNSW) Nov, 0

5 4..3 Mchine voltge eqution for ech phe The voltge eqution of phe of the mchine i thu given by v d di d f Ri Ri L (4..) dt dt dt where R i the reitnce of ech winding. Note tht i b nd i c re not preent in the voltge eqution for phe, even though they contribute to the totl flux linkge of winding. Similrly, for the other two ttor phe, d di d v Ri Ri L dt dt dt b b bf b b b (4..3) v c d di d c c cf Ric Ric L (4..4) dt dt dt The Bck-emf of ech phe widing We now ume tht the rotor flux ditribution i coinuoidl round the rotor pole. (n other word, inuoidl ditributed rotor winding i being umed). When the rotor rotte t peed yn, the ngulr poition (in electricl rdin) of the rotor t ny time t i p dt dt yn in elec rdin (4..5) where i the rbitrry ngle (electricl) of the rotor t t = 0 nd f rd/ec (electricl). The flux linkge with winding due to the rotor flux i Note tht L f = L bf = L cf = L m. co t L co t (4..6) f f m f The voltge induced in winding i df ef Lm f int (4..7) dt Lm f cot (4..8) v 90 e f f Figure 4..6 Phe flux linkge nd bck-emf wveform Section 4. Syn Motor repreenttion 5 F. Rhmn (EET, UNSW) Nov, 0

6 At contnt peed, the flux linkge wveform (phor) thu lg the induced voltge wveform (phor) by 90 electricl degree indicted by figure 4..6 nd 4..7 repectively. E f f Figure Equivlent circuit nd phor digrm With negligible ttor reitnce R, the AC circuit repreenttion of ech phe nd it phor digrm re indicted in figure 4..8 nd 4..9 repectively. Note tht in the tedy-tte, the rotor rotte t ynchronou peed, f yn mech rd/ec nd = f i in elec rd/ec. p R 0; Lo-le SM under no lod jx =j L V0 E f Figure 4..8 Phor digrm t contnt frequency (peed) V jx E f jx = 0 E f or = 0 or V f f () Under-excited rotor (b) Over-excited rotor Figure 4..9 f the ttor reitnce R i negligibly mll nd ll other loe re neglected, the current phor drwn from the upply for n unloded motor mut be t right ngle to V or E f phor, o to reflect zero input nd developed power (Vco nd E f co, repectively). Thi ume tht the iron loe re lo zero. The phe current i then given by Section 4. Syn Motor repreenttion 6 F. Rhmn (EET, UNSW) Nov, 0

7 0 Ef V jx (4..9) Note tht the umption of no lo require tht mut lo be zero. When the rotor i underexcited o tht E f < V, would lg the V or E f phor by 90. The phor would led the V or E f phor by 90 when the rotor i overexcited i.e., when E f > V. Thee two ce re depicted in the phor digrm below. Obviouly, for the lole nd unloded motor, will be zero when V = E f. For non idel motor with ome loe, the ngle will not be zero for the un-loded motor nd the current phor will be minimum but nonzero when V equl E f. The current phor will then led V nd E f for overexcittion or lg V nd E f for underexcittion. For low lo mchine, the phe ngle between V nd will be lightly mller thn 90. The ngle will be mll but non zero. When the motor i loded through the hft, the ngle of will fll further nd the ngle will incree ccordingley. Such n operting condition i depicted in the phor digrm of figure 4.. for underexcited (Figure 4..) nd overexcited (Figure 4..b) rotor. Note tht by djiting the rotor excittion, the mgnitude of E f cn be vried, nd the motor cn be mde to operte t ny power fctor ngle, lgging or leding R not negligible; SM on lod R jx =j L V0 E f V jx E f R f Figure 4..0 jx V R E f f () Under-excited motor, R 0 (b) Over-excited motor, R 0 Figure 4.. Section 4. Syn Motor repreenttion 7 F. Rhmn (EET, UNSW) Nov, 0

8 4..5 Stedy-tte lod chrcteritic When the mchine i loded through the hft, the motor will tke rel power. The rotor will then fll behind the ttor (revolving) field. From the circuit digrm of figure 4..0, the motor current i given by V0 E f V0 E f (4..0) R jx Z where Z R X ; X L (4..) X nd tn. (4..) R f R i negiligible, o tht Z X nd, V E (4..3) X X The rel prt of (i.e., in-phe with V) i given by Re E X f in (4..4) The developed power i given by VE f P V Re in Wtt/phe (4..5) X Thi power (per phe) given by 4..5 i poitive when i negtive, ie when the rotor pole (field) lg the ttor pole (field) by ngle.. Thu. the mchine ct motor, when < 0, nd it ct genertor, when > 0. Thi i depicted in figure The developed torque i given by T P P f p Nm/phe yn 3p VE X f in Nm (4..6) The negtive ign h been dropped, uming tht for poitive (motoring) torque, negtive i implied. Section 4. Syn Motor repreenttion 8 F. Rhmn (EET, UNSW) Nov, 0

9 P, Wtt T, Nm Motor Genertor 4..6 Phor digrm nd reference frme Figure 4.. Note tht t contnt peed, the flux linkge, pplied nd induced voltge nd current re ll inuoidl quntitie of the me frequency. All of thee quntitie cn be preented in ingle phor digrm indicted in figure j q X q-xi j d X V jx q E f d-xi d f Figure 4..3 The phe current cn be thought of hving two component: d producing n mmf long the d-xi (or pole xi) nd q producing n mmf long the q-xi (inter-pole xi). (Thi men tht the q component of produce flux-linkge phor which i in qudture (rther thn in phe) with f ). The vector um of thee two field (which re 90 o prt) ccount for the net mmf produced by the ttor current. Thu, when lg Ef, i.e. when the rotor i under-excited, d tend to mgnetie the rotor. d tend to demgnetie the rotor when led Ef. Reolving into d nd q in thi wy help to nlye the lient-pole (non cylndricl) ynchronou motor, which i nlyed in the following ection. t will lo explin the elfynchronou drive principle which i decribed in ltter ection. The d nd q phor in figure 4..3 re inuoidl quntitie t ttor upply frequency. Thee re then inuoidl current (one ine nd one coine function of time) flowing in two fictitiou winding locted in the ttor, which develop mmf long the rotor d- nd q- xe. ( d nd q could lo be expreed in the rotor reference frme, which i reference frme firmly ttched to the rotor. n thi ce, d nd q re current in two fictitiou winding ttched long the d- nd q- xe of the rotor. n thi reference frme, d nd q become DC quntitie when the peed of rottion i contnt). Section 4. Syn Motor repreenttion 9 F. Rhmn (EET, UNSW) Nov, 0

10 4..7 Slient-Pole (non-uniform irgp) ynchronou motor The lient-pole mchine doe not hve n uniform irgp. The irgp length i hort long the pole xi nd long long the inter-pole xi, indicted in figure 4..4(). Conequently, the fluxe produced by the ttor current vry ccording to the ngulr poition of the rotor it rotte. Reolving the ttor current (of ech phe) into d- nd q- xe component, it i cler from figure 4..4() tht d, cting long the pole xi (or d-xi) of the rotor, produce more flux per mpere of current long thi xi, thn q doe long the q-xi. The different contnt of proportionlity (L d nd L q ) between flux nd current long the two xe implie tht the ynchronou rectnce long the two xe re different. (n fct the ynchronou rectnce of thi mchine chnge continuouly with rotor poition but it mximum nd minimum vlue for inuoidlly ditributed rotor flux will uffice for nlyi). We only ume tht the ttor nd rotor flux ditribution re inuoidl in pce. Thu, ynchronou rectnce of the fictitiou d- xi winding X d = L d (4..7) Similrly, the ynchronou rectnce of the q-xi fictitiou winding i X q = L q (4..8) n generl, X d > X q Phor digrm of the Slient-Pole Motor j q X q q- xi j d X d V q E f d- xi d () (b) Figure 4..4 Note tht the phor digrm of figure 4..4(b) i bed on the umption tht R = 0. Noting lo tht i negtive for motoring nd neglecting R, voltge long the d- nd q- xe cn be written X q q V in (4..9) nd E f X V co (4..30) d d Section 4. Syn Motor repreenttion 0 F. Rhmn (EET, UNSW) Nov, 0

11 The power input to the motor (which i lo the developed power ince other loe hve been neglected) i P V co V in W/phe (4..3) q d Solving for d nd q from (4..9) nd (4..30) nd ubtituting into (4..3) VE f V X d X q P in in Xd XdX q W/phe (4..3) nd the developed torque i given by T 3P yn Nm 3p VE f V X d X q in in Nm Xd XdX q (4..33) Agin, it i implied tht i negtive for poitive motoring torque. The econd term in the torque expreion of Eq i minly due to the liency of the rotor. High liency, i.e., lrge X d X q, hould be deirble, for obtining mximum torque in the tedy-tte. Note tht for non-lient-pole motor, X d = X q, nd the econd term in the torque expreion then vnihe. The econd term i independent of rotor excittion nd it i clled reluctnce torque. Figure 4..5 how the torque - chrcteritic of lient pole motor for vriou excittion level E f /V. E f /V = T, Nm Figure 4..5 Note tht the mximum torque, T mx for thi motor occur for < 90. Section 4. Syn Motor repreenttion F. Rhmn (EET, UNSW) Nov, 0

12 4..8 Permnent Mgnet Synchronou Motor Rotor excittion, E f, i fixed. E f i determined by the mgnetic circuit, (mteril, dimenion, rotor deign, effective irgp etc). Mgnet mteril h permebility cloe to tht of free pce. The direct xi rectnce, X d, i uully mller thn X q. The mgnetic circuit i normlly o deigned tht the motor operte with ner unity power fctor t full lod. d- xi q- xi d- xi q- xi Figure 4..6 Rotor cro ection of interior permnent mgnet ynchronou motor Totl torque Torque due to permnent mgnet T, Nm Reluctnce torque Figure Torque v ngle chrcteritic of the PM motor of figure 4..6() 4..9 The Synchronou Reluctnce Motor The ynchronou reluctnce motor doe not hve ny electro- or permnent mgnet in the rotor. n other word, the rotor doe not hve ny excittion. Figure 4..8 how the cro ection of the rotor of ynchronou reluctnce motor. n the bence of ny rotor excittion, ie for E f = 0, thi mchine cn lo operte ynchronou motor, provided tht X d i not equl to X q. The econd term in eqution would then be olely reponible for the motoring torque. Thu, for E f = 0 nd X d > X q, the motor i ble to produce net poitive torque while running t Section 4. Syn Motor repreenttion F. Rhmn (EET, UNSW) Nov, 0

13 ycnhronou peed. Thi i the principle of the ynchronou reluctnce (Synchrel) motor. Figure 4..8 how the torque v chrcteritic of ynchronou reluctnce motor. Obviouly, the fctor X d /X q (the liency rtio) i n importnt fctor for producing high torque in the tedy-tte nd lo for chieving dynmic repone of thi motor. The developed torque for thi motor i given by T 3pV Xd Xq in XdX q Nm (4..34) Nrrow iron bridge Q-xi D-xi Air flux Brrier Shft Figure Cro ection of ynchronou reluctnce motor. T, Nm Figure 4..9 T- chrcteritic of the Synchrel motor. Section 4. Syn Motor repreenttion 3 F. Rhmn (EET, UNSW) Nov, 0