Preliminary Fundamentals

Transcription

1 1.0 Introduction Prliminary Fundamntals In all of our prvious work, w assumd a vry simpl modl of th lctromagntic torqu T (or powr) that is rquird in th swing quation to obtain th acclrating torqu. This simpl modl was basd on th assumption that thr ar no dynamics associatd with th machin intrnal voltag. This is not tru. W now want to construct a modl that will account for ths dynamics. To do so, w first nd to nsur that w hav adquat background rgarding prliminary fundamntals, which includ som ssntial lctromagntic thory, and basics of synchronous machin construction & opration. 2.0 Som ssntial lctromagntic thory 2.1 Slf inductanc Slf inductanc indicats th magnitud of th magntic coupling btwn a circuit and itslf. It is givn, with units of hnris, by 1

2 L i 1 (1) W s that th slf-inductanc L 11 is th ratio of th flux φ 11 from coil 1 linking with coil 1, λ 11 to th currnt in coil 1, i 1. Sinc th flux linkag λ 11 is th flux φ 11 linking with coil 1, and sinc this flux links onc pr turn, and sinc th numbr of turns is N 1, thn 2.2 Faraday s law N (2) Any chang of flux linkags sn by a circuit inducs a voltag in that circuit. Th inducd voltag is givn by d d( Li) dt dt (3) whr (3) accounts for th cas of tim variation in L, i, or both. If L dos not vary with tim, thn d di L dt dt (4) 2

3 2.3 Mutual inductanc For a pair of circuits, th mutual inductanc L 12 is L 12 i 12 2 (5) W obsrv that L 12 is th ratio of th flux from coil 2 linking with coil 1, λ 12 to th currnt is coil 2, i 2. Mor gnrally, for a group of circuits labld 1, 2,, w s that L 1 L 2 i 11 1 i 21 1 L 12 L i 22 2 i (6) Hr, L 11, L 22, ar slf inductancs, and L 12, L 21, ar mutual inductancs. From (6), w s a mor gnral dfinition of slf and mutual inductancs, according to: L ii i i (8) i 3

4 L ij i i j (9) In th cas of slf inductanc, bcaus λ i is producd by i i thir dirctionalitis will always b consistnt such that currnt incrass produc flux linkag incrass. Thrfor L ii is always positiv. In th cas of mutual inductanc, whthr currnt incrass in on circuit produc flux linkag incrass in th othr circuit dpnds on th dirctionality of th currnts and fluxs. Th rul w will us is this: L ij is positiv if positiv currnts in th two circuits produc slf and mutual fluxs in th sam dirction. 2.4 Inductanc and magntic circuits W dfin magntomotiv forc (MMF), as th forc that rsults from a currnt i flowing in N turns of a conductor. W will dnot it with F, xprssd by: MMF F Ni (10) If th conductor is wound around a magntic circuit having rluctanc R, thn th MMF will caus flux to flow in th magntic circuit according to 4

5 F Ni R R (11) If th cross-sctional ara A and prmability μ of th magntic circuit is constant throughout, thn R l A (12a) whr l is th man lngth of th magntic circuit. Th prmanc is givn by P 1 R (12b) Magntic circuit rlations dscribd abov ar analogous to Ohm s law for standard circuits, in th following way: FV, φi, RR, PY (13) So that Th F hr should b F. F V I R R (14) W also show in th appndix (s qs (A8), (A9a)) that L NN L R N 2 1 L11 R (15) 5

6 2.5 Constant flux linkag thorm Considr any closd circuit having finit rsistanc flux linkag du to any caus whatsovr othr mf s not du to chang in λ no sris capacitanc Thn d dt ri (16) W know that flux linkags can chang, and (16) tlls us how: whnvr th balanc btwn th mfs and th rsistanc drops bcom non-zro, i.., d dt ri (17) But, can thy chang instantly, i.., can a crtain flux linkag λ chang from 4 to 5 wbr-turns in 0 sconds? To answr this qustion, considr intgrating (16) with rspct to tim t from t=0 to t= t. W obtain 6

7 t r idt 0 Trm1 t d dt dt 0 Trm2 t dt 0 Trm3 (18) Notic that ths trms ar, for th intrval 0 t, Trm 1: Th ara undr th curv of i(t) vs. t Trm 2: Th ara undr th curv of dλ/dt vs. t, which is λ( t) (rad dlta lambda of dlta t ). Trm 3: Th ara undr th curv of (t) vs. t. Now w know that w can gt an instantanous (stp) chang in currnt short th circuit or opn th circuit, and w know that w can gt an instantanous (stp) chang in voltag opn/clos a switch to insrt a voltag sourc into th circuit. And so i(t) and/or (t) may chang instantanously in (18). But considr applying th limit as Δt0 to (18). In this cas, w hav: 7

8 t lim r idt t0 0 Trm1 lim ( t) t0 Trm2 t lim dt t0 0 Trm3 Ths should b lim as Δt0 (and not lim as t0). (19) Evn with a stp chang in i(t) or (t), thir intgrals will b zro in th limit. Thrfor w hav: This should b lim as Δt0 (and not lim as t0). 0 lim ( t) t 0 Trm2 0 (20) This implication of (20) is that th flux linkags cannot chang instantanously. This is th constant-flux-linkag thorm (CFLT). CFLT: In any closd lctric circuit, th flux linkags will rmain constant immdiatly aftr any chang in Th currnt Th voltag Th position of othr circuits to which th circuit is magntically coupld. 8

9 Th CFLT is particularly usful whn L ii or L ij of a circuit changs quickly. It allows us to assum λ stays constant so that w can obtain currnts aftr th chang as a function of currnts bfor th chang. 3.0 Basics of synchronous machins 2.1 Basic construction issus In this sction, w prsnt only th vry basics of th physical attributs of a synchronous machin. W will go into mor dtail rgarding windings and modling latr. Th synchronous gnrator convrts mchanical nrgy from th turbin into lctrical nrgy. Th turbin convrts som kind of nrgy (stam, watr, wind) into mchanical nrgy, as illustratd in Fig. 1 [i]. 9

10 Fig. 1 [i] Th synchronous gnrator has two parts: Stator: carris 3 (3-phas) armatur windings, AC, physically displacd from ach othr by 120 dgrs Rotor: carris fild windings, connctd to an xtrnal DC sourc via slip rings and brushs or to a rvolving DC sourc via a spcial brushlss configuration. Fig. 2 shows a simplifid diagram illustrating th slip-ring connction to th fild winding. 10

11 Stator Rotor winding Brushs +- Stator winding Slip rings Fig. 2 Fig. 3 shows th rotor from a 200 MW stam gnrator. This is a smooth rotor. Fig. 3 11

12 Fig. 4 shows th rotor and stator of a hydro-gnrator, which uss a salint pol rotor. Fig. 4 Fig. 5 illustrats th synchronous gnrator construction for a salint pol machin, with 2 pols. Not that Fig. 5 only rprsnts on sid of ach phas, so as to not crowd th pictur too much. In othr words, w should also draw th Phas A rturn conductor 180 away from th Phas A conductor shown in th pictur. Likwis for Phass B and C. 12

13 ROTOR (fild winding) + Phas A N STATOR (armatur winding) Phas B DC Voltag + Th ngativ trminal for ach phas is 180 dgrs from th corrsponding positiv trminal. + Phas C S Fig. 5 A Two Pol Machin (p=2) Salint Pol Structur Fig. 6 shows just th rotor and stator (but without stator winding) for a salint pol machin with 4 pols. S N S A Four Pol Machin (p=4) (Salint Pol Structur) N Fig. 6 13

14 Th diffrnc btwn smooth rotor construction and salint pol rotor construction is illustratd in Fig. 7. Not th air-gap in Fig. 7. Air-gap Fig. 7 W dfin synchronous spd as th spd for which th inducd voltag in th armatur (stator) windings is synchronizd with (has sam frquncy as) th ntwork voltag. Dnot this as ω R. In North Amrica, In Europ, ω R =2π(60)= rad/sc ω R =2π(50)= rad/sc 14

15 On an airplan, ω R =2π(400)= rad/sc Th mchanical spd of th rotor is rlatd to th synchronous spd through: 2 m p (21) whr both ω m and ω ar givn in rad/sc. This may b asir to think of if w writ 2 p m (22) Thus w s that, whn p=2, w gt on lctric cycl for vry on mchanical cycl. Whn p=4, w gt two lctrical cycls for vry on mchanical cycl. If w considr that ω R must b constant from on machin to anothr, thn machins with mor pols must rotat mor slowly than machins with lss. It is common to xprss ω mr in RPM, dnotd by N; w may asily driv th convrsion from analysis of units: 15

16 N mr =(ω m rad/sc)*(1 rv/2π rad)*(60sc/min) = (30/π)ω mr Substitution of ω mr =(2/p) ω R =(2/p)2πf=4πf/p N mr = (30/π)(4πf/p)=120f/p (23) Using (3), w can s variation of N mr with p for f=60 Hz, in Tabl 1. Tabl 1 No. of Pols (p) Synchronous spd (NmR) Bcaus stam-turbins ar abl to achiv high spds, and bcaus opration is mor fficint at thos spds, most stam turbins ar 2 pol, oprating at 3600 RPM. 16

17 At this rotational spd, th surfac spd of a 3.5 ft diamtr rotor is about 450 mil/hour. Salint pols incur vry high mchanical strss and windag losss at this spd and thrfor cannot b usd. All stam-turbins us smooth rotor construction. Bcaus hydro-turbins cannot achiv high spds, thy must us a highr numbr of pols,.g., 24 and 32 pol hydro-machins ar common. But bcaus salint pol construction is lss xpnsiv, all hydro-machins us salint pol construction. Fig. 8 illustrats svral diffrnt constructions for smooth and salint-pol rotors. Th rd arrows indicat th dirction of th flux producd by th fild windings. 17

18 Synchronous gnrator Rotor construction Round Rotor Salint Pol Two pol s = 3600 rpm Four Pol s = 1800 rpm Eight Pol s = 900 rpm Fig. 8 Th synchronous machin typically has two sparat control systms th spd govrning systm and th xcitation systm, as illustratd in Fig. 9 blow. Our main intrst in this cours is synchronous machin modling. W will only touch on a fw issus rlatd to th control systms. 18

19 Fig Rotating magntic fild Th following outlins th concptual stps associatd with production of powr in a synchronous gnrator. 1. DC is supplid to th fild winding. 2. If th rotor is stationary, th fild winding producs magntic flux which is strongst radiating outwards from th cntr of th pol fac and diminishs with distanc along th air-gap away from th pol fac cntr. Figur 10 illustrats. Th lft-hand-figur plots flux dnsity as a function of angl from th main axis. Th right-hand plot shows th main axis and th lins of flux. Th angl θ masurs th point on th stator from th main axis, which is th a-phas axis. In this particular cas, w hav alignd th main axis with th dirct-axis of th rotor. 19

20 θ Dirct rotor axis Stator 0 θ B, flux dnsity in th air gap Air gap Rotor Magntic fild lins Fig Th turbin rotats th rotor. This producs a rotating magntic fild (or a sinusoidal travling wav) in th air gap, i.., th plot on th lft of Fig. 10 movs with tim. Figur 11 illustrats, whr w s that, for fixd tim (just on of th plots), thr is sinusoidal variation of flux dnsity with spac. Also, if w stand on a singl point on th stator (.g., θ=90 ) and masur B as a function of tim, w s that for fixd spac (th vrtical dottd lin at 90, and th rd y on th picturs to th right), thr is sinusoidal variation of flux dnsity w/tim. 20

21 0 θ θ N θ θ N θ θ N θ=90 Fig Givn that th stator windings, which run down th stator sids paralll to th lngth of th gnrator ar fixd on th stator (lik th y of Fig. 11), thos conductors will s a tim varying flux. Thus, by Faraday s law, a voltag will b inducd in thos conductors. a. Bcaus th phas windings ar spatially displacd by 120, thn w will gt voltags that ar timdisplacd by 120. b. If th gnrator trminals ar opn-circuitd, thn th amplitud of th voltags ar proportional to Spd 21

22 Magntic fild strngth And our story nds hr if gnrator trminals ar opn-circuitd. 5. If, howvr, th phas (armatur) windings ar connctd across a load, thn currnt will flow in ach on of thm. Each on of ths currnts will in turn produc a magntic fild. So thr will b 4 magntic filds in th air gap. On from th rotating DC fild winding, and on ach from th thr stationary AC phas windings. 6. Th thr magntic filds from th armatur windings will ach produc flux dnsitis, and th composition of ths thr flux dnsitis rsult in a singl rotating magntic fild in th air gap. W dvlop this hr. Considr th thr phas currnts: i I cos t i i a b c I cos( t I cos( t 120) 240) (24) Now, whnvr you hav a currnt carrying coil, it will produc a magntomotiv forc (MMF) qual to Ni. And so ach of th abov thr currnts produc a tim varying MMF around th stator. Each MMF will hav a 22

23 maximum in spac, occurring on th axis of th phas, of F am, F bm, F cm, xprssd as F F F am bm cm ( t) ( t) ( t) F F F m m m cos t cos( t cos( t 120) 240) (25) Rcall that th angl θ is masurd from th a-phas axis, and considr points in th airgap. At any tim t, th spatial maximums xprssd abov occur on th axs of th corrsponding phass and vary sinusoidally with θ around th air gap. W can combin th tim variation with th spatial variation in th following way: F (, t) F a F (, t) F b F (, t) F c am bm cm ( t)cos ( t)cos( 120) ( t)cos( 240) (26) Not ach individual phas MMF in (26) varis with θ around th air gap and has an amplitud that varis with tim. Substitution of (25) into (26) yilds: 23

24 F a (, t) F (, t) b F (, t) c F F F m m m cos t cos cos( t cos( t 120)cos( 120) 240)cos( 240) (27) Now do th following: Add th thr MMFs in (27): F(, t) F (, t) F (, t) F F F F m m m a cos t cos cos( t cos( t b (, t) 120)cos( 120) 240)cos( 240) c (28) Us cosαcosβ=0.5[cos(α-β)+cos(α+β)] and thn simplify, and you will obtain: 3 F(, t) Fm cos( t ) 2 (29) Equation (29) charactrizs a rotating magntic fild, just as in Fig This rotating magntic fild from th armatur will hav th sam spd as th rotating magntic fild from th rotor, i.., ths two rotating magntic filds ar in synchronism. 8. Th two rotating magntic filds, that from th rotor and th composit fild from th armatur, ar lockd 24

25 in, and as long as thy rotat in synchronism, a torqu (Torqu=P/ω m =Forc radius, whr Forc is tangntial to th rotor surfac), is dvlopd. This torqu is idntical to that which would b dvlopd if two magntic bars wr fixd on th sam pivot [ii, pg. 171] as shown in Fig 3. In th cas of synchronous gnrator opration, w can think of bar A (th rotor fild) as pushing bar B (th armatur fild), as in Fig. 12a. In th cas of synchronous motor opration, w can think of bar B (th armatur fild) as pulling bar A (th rotor fild), as in Fig. 12b. S S S N N Bar A Bar B N S Bar A Bar B N Fig 12a: Gnrator opration Fig 12b: Motor opration Fig

26 Appndix: Mutual inductanc Lt s considr anothr arrangmnt as shown in Fig. A1 blow. i 1 φ i 2 N 1 N 2 Fig. A1 W hav for ach coil: L L i 11 (A1) 1 i 22 (A2) 2 W can also dfin L 12 and L 21. L 12 is th ratio of That is, th flux from coil 2 linking with coil 1, λ 12 to th currnt in coil 2, i 2. L 12 i 12 (A3) 2 26

27 whr th first subscript, 1 in this cas, indicats links with coil 1 and th scond subscript, 2 in this cas, indicats flux from coil 2. Hr, w also hav that 12 N 1 12 N112 L12 (A4) i2 Likwis, w hav that L i 1 (A5a) 21 N 2 21 N221 L21 (A5b) i1 Now lt s assum that all flux producd by ach coil links with th othr coil. Th implication of this is that thr is no lakag flux, as illustratd in Fig. A2. This lakag flux is assumd to b zro. i 1 φ i 2 N 1 N 2 Fig. A2 Although in rality thr is som lakag flux, it is quit small bcaus th iron has much lss rluctanc than th air. With this assumption, thn w can writ: th flux from coil 2 linking with coil 1 is qual to th flux from coil 2 linking with coil 2, i.., 27

28 A l N2i2 (A6a) th flux from coil 1 linking with coil 2 is qual to th flux from coil 1 linking with coil 1, i.., A N (A6b) l i1 Substitution of (A6a) and (A6b) into (A4) and (A5b), rspctivly, rsults in: L L A N N i N l A N N (A7a) R N1N2 i2 i2 l A N N i N l A N N (A7b) R N2N1 i1 i1 l Examination of (A7a) and (A7b) lads to L NN L12 (A8) R Also rcall 2 N L R or in subscriptd notation N 2 1 L11 R (A9a) 2 N2 L (A9b) R 22 Solving for N 1 and N 2 in (A9a) and (A9b) rsults in N L R (A10a)

29 N L R (A10b) 2 22 Now substitut (A10a) and (A10b) into (A8) to obtain L R L R L L L L R (21) Dfinition: L 12 =L 21 is th mutual inductanc and is oftn dnotd M. Mutual inductanc givs th ratio of flux from coil k linking with coil j, λ jk That is, to th currnt in coil k, i k, M 12 i2 21 i1 [ i ] [ ii ] A. Fitzgrald, C. Kingsly, and A. Kusko, Elctric Machinry, Procsss, Dvics, and Systms of Elctromchanical Enrgy Convrsion, 3rd dition, 1971, McGraw Hill. 29