COURSE: INTRODUCTION TO ELECTRICAL MACHINES PART 3. Prof Elisete Ternes Pereira, PhD INTRODUCTION TO ROTATING MACHINES

Transcription

1 COURSE: ELECTRICAL MACHINES PART 3 Prf Elisete Ternes Pereira, PhD ROTATING MACHINES

2 ROTATING MACHINES INTRODUCTION Electrmagnetic energy cnversin ccurs when changes in the flux linkage result frm mechanical mtin. V ind dλ = dt Schematic view f a simple, tw-ple, single-phase synchrnus generatr. In rtating machines, vltages are generated in windings r grups f cils by: Rtating these windings mechanically thrugh a magnetic field, Mechanically rtating a magnetic field past the winding, designing the magnetic circuit s that the reluctance varies with rtatin f the rtr. By any f these methds, the flux linking a specific cil is changed cyclically, and a time-varying vltage is generated. ROTATING MACHINES INTRODUCTION A set f such cils cnnected tgether is typically referred t as an armature winding. In general, the term armature winding is used t refer t a winding n a rtating machine which carry ac currents. In ac machines such as synchrnus r inductin machines, the armature winding is typically n the statinary prtin f the mtr referred t as the statr, in which case these windings may als be referred t as statr windings. In a dc machine, the armature winding is fund n the rtating member, referred t as the rtr. the armature winding f a dc machine cnsists f many cils cnnected tgether t frm a clsed lp. Simple, tw-ple, single-phase synchrnus generatr.

3 ROTATING MACHINES INTRODUCTION In sme machines, such as variable reluctance machines and stepper mtrs, there are n windings n the rtr. Operatin f these machines depends n the nnunifrmity f air-gap reluctance assciated with variatins in rtr psitin in cnjunctin with time-varying currents applied t their statr windings. Rtating electric machines take many frms and are knwn by many names: dc, synchrnus, permanent-magnet, inductin, variable reluctance, hysteresis, brushless, and s n. Althugh these machines appear t be quite dissimilar, their behavir are quite similar, and it is ften helpful t think f them in terms f the same physical picture. ROTATING MACHINES AC & DC MACHINES - INTRODUCTION

4 AC MACHINES Traditinal ac machines fall int ne f tw categries: Synchrnus machines: rtr-winding currents are supplied directly frm the statinary frame thrugh a rtating cntact. Inductin machines: rtr currents are induced in the rtr windings by a cmbinatin f the time-variatin f the statr currents and the mtin f the rtr relative t the statr. Synchrnus Machines Synchrnus machines: Cnsider the very much simplified salient-ple ac synchrnus generatr: The field-winding f this machine prduces a single pair f magnetic ples (similar t that f a bar magnet), and hence this machine is referred t as a tw-ple machine. With rare exceptins, the armature winding f a synchrnus machine is n the statr, and the field winding is n the rtr, The field winding is excited by direct current cnducted t it by means f statinary carbn brushes which cntact rtating slip rings r cllectr rings.

5 Synchrnus Machines The armature winding, cnsisting here f nly a single cil f N turns, is indicated in crss sectin by the tw cil sides a and -a. The cnductrs frming these cil sides are parallel t the shaft f the machine and are cnnected in series by end cnnectins (nt shwn in the figure). The rtr is turned at a cnstant speed by a surce f mechanical pwer cnnected t its shaft. The armature winding is assumed t be pen-circuited and hence the flux in this machine is prduced by the field winding alne. Flux paths are shwn schematically by dashed lines. Synchrnus Machines An idealized analysis wuld assume a sinusidal distributin f magnetic flux in the air gap. The resultant radial distributin f air-gap flux density B is shwn in the figure bellw as a functin f the spatial angle θ a (measured with respect t the magnetic axis f the armature winding) arund the rtr periphery. (a) Space distributin f flux density and (b) crrespnding wavefrm f the generated vltage fr the single-phase generatr

6 Synchrnus Machines As the rtr rtates, the flux-linkages f the armature winding change with time. Under the assumptin f a sinusidal flux distributin and cnstant rtr speed, the resulting cil vltage will be sinusidal in time. The cil vltage passes thrugh a cmplete cycle fr each revlutin f the tw-ple machine. Its frequency in cycles per secnd (Hz) is the same as the speed f the rtr in revlutins per secnd: the electric frequency f the generated vltage is synchrnized with the mechanical speed, and this is the reasn fr the designatin "synchrnus" machine. Thus a tw-ple synchrnus machine must revlve at 3600 revlutins per minute t prduce a 60-Hz vltage. Synchrnus Machines A great many synchrnus machines have mre than tw ples. The figure shws a fur-ple single-phase generatr. The field cils are cnnected s that the ples are f alternate plarity. There are tw cmplete wavelengths, r cycles, in the flux distributin arund the periphery. The armature winding nw cnsists f tw cils a1, -a1 and a2, -a2 cnnected in series. The span f each cil is ne wavelength f flux. The generated vltage nw ges thrugh tw cmplete cycles per revlutin f the rtr. The frequency in hertz will be twice the speed in revlutins per secnd.

7 Synchrnus Machines When a machine has mre than tw ples, it is cnvenient t cncentrate n a single pair f ples and t recgnize that the electric, magnetic, and mechanical cnditins assciated with every ther ple pair are repetitins f thse fr the pair under cnsideratin. Fr this reasn it is cnvenient t express angles in electrical degrees r electrical radians rather than in physical units. One pair f ples in a multiple machine r ne cycle f flux distributin equals 360 electrical degrees r 2π electrical radians. Since there are (ples/2) cmplete wavelengths, r cycles, in ne cmplete revlutin, it fllws that: ple θ θ ae = 2 Where: θ ae angle in electrical units θ a spatial angle. This same relatinship applies t all angular measurements in a multiple machine; their values in electrical units will be equal t (ples/2) times their actual spatial values. a Synchrnus Machines The cil vltage f a multiple machine passes thrugh a cmplete cycle every (ples/2) times each revlutin. The electrical frequency f e f the vltage generated in a synchrnus machine is therefre: ple n f e = 2 60 Hz Where: n the mechanical speed in revlutins per minute, and hence n/60 is the speed in revlutins per secnd. The electrical frequency f the generated vltage in radians per secnd is: ple ω = 2 e ω m rad Where: ω m is the mechanical speed in radians per secnd. / sec

8 Synchrnus Machines The rtrs in the 2 figures belw have salient, r prjecting, ples with cncentrated windings. Figure belw shws a nnsalient-ple, r cylindrical rtr. The field winding is a tw-ple distributed winding; the cil sides are distributed in multiple slts arund the rtr periphery and arranged t prduce an apprximately sinusidal distributin f radial air-gap flux. Synchrnus Machines ple n f e = 2 60 Hz A salient-ple cnstructin is characteristic f hydrelectric generatrs because hydraulic turbines perate at relatively lw speeds, and hence a relatively large number f ples is required t prduce the desired frequency; the salient-ple cnstructin is better adapted mechanically t this situatin. Steam turbines and gas turbines, hwever, perate best at relatively high speeds, and turbine-driven alternatrs r turbine generatrs are cmmnly tw- r fur-ple cylindrical-rtr rtr machines.

9 Synchrnus Machines Mst f the wrld's pwer systems are three-phase systems and, as a result, with very few exceptins, synchrnus generatrs are three-phase machines. Fr the prductin f a set f three vltages phase-displaced by 120 electrical degrees in time, a minimum f three cils phase-displaced 120 electrical degrees in space must be used. A simplified schematic f a three-phase, tw-ple machine with ne cil per phase: The three phases are designated by the letters a, b, and c. Synchrnus Machines In an elementary three-phase, fur-ple machine, a minimum f tw such sets f cils must be used, as illustrated; in an elementary multiple machine, the minimum number f cils sets is given by ne half the number f ples. The tw cils in each phase are cnnected in series s that their vltages add, and the three phases may then be either Y- r Δ-cnnected.

10 Synchrnus Machines The figure shws hw the cils may be intercnnected t frm a Y cnnectin. Since the vltages in the cils f each phase are identical, a parallel cnnectin is als pssible, e.g., cil (a, -a) in parallel with cil (a', -a'), and s n. Synchrnus Machines When a synchrnus generatr supplies electric pwer t a lad, the armature current creates a magnetic flux wave in the air gap which rtates at synchrnus speed, This flux reacts with the flux created by the field current, and electrmechanical trque results frm the tendency f these tw magnetic fields t align. In a generatr this trque ppses rtatin, and mechanical trque must be applied frm the prime mver t sustain rtatin. This electrmechanical trque is the mechanism thrugh which the synchrnus generatr cnverts mechanical t electric energy.

11 Synchrnus Machines The cunterpart f the synchrnus generatr is the synchrnus mtr. A cutaway view f a three-phase, 60-Hz synchrnus mtr is shwn in the figure: Alternating current is supplied t the armature winding n the statr, and dc excitatin is supplied t the field winding n the rtr. The magnetic field prduced by the armature currents rtates at synchrnus speed. T prduce a steady electrmechanical trque, the magnetic fields f the statr and rtr must be cnstant in amplitude and statinary with respect t each ther. Synchrnus Machines In a synchrnus mtr, the steady-state speed is determined by the number f ples and the frequency f the armature current. Thus a synchrnus mtr perated frm a cnstantfrequency ac surce will perate at a cnstant steady-state speed. In a mtr the electrmechanical trque is in the directin f rtatin and balances the ppsing trque required t drive the mechanical lad. The flux prduced by currents in the armature f a synchrnus mtr rtates ahead f that prduced by the field, thus pulling n the field (and hence n the rtr) and ding wrk. This is the ppsite f the situatin in a synchrnus generatr, where the field des wrk as its flux pulls n that f the armature, which is lagging behind. In bth generatrs and mtrs, an electrmechanical trque and a rtatinal vltage are prduced.

12 Inductin Machines In the Inductin Machines, like the synchrnus machine, the statr winding is excited with alternating currents. But, Rtr currents are prduced by inductin, i.e., transfrmer actin. The inductin machine may be regarded as a generalized transfrmer in which electric pwer is transfrmed between rtr and statr tgether with a change f frequency and a flw f mechanical pwer. In cntrast t a synchrnus machine in which a field winding n the rtr is excited with dc current, alternating currents flw in the rtr windings f an inductin machine. In inductin machines, alternating currents are applied directly t the statr windings. Inductin Machines Althugh the inductin mtr is the mst cmmn f all mtrs, it is seldm used as a generatr; Its perfrmance characteristics as a generatr are unsatisfactry fr mst applicatins, althugh in recent years it has been fund t be well suited fr wind-pwer applicatins. The inductin machine may als be used as a frequency changer.

13 Inductin Machines In the inductin mtr, the rtr windings are electrically shrt-circuited and frequently have n external cnnectins; currents are induced by transfrmer actin frm the statr winding. A squirrel-cage inductin mtr is shwn: Here the rtr "windings" are actually slid aluminum bars which are cast int the slts in the rtr and which are shrted tgether by cast aluminum rings at each end f the rtr. This type f rtr cnstructin results in inductin mtrs which are relatively inexpensive and highly reliable, factrs cntributing t their immense ppularity and widespread applicatin. Inductin Machines As in a synchrnus mtr, the armature flux in the inductin mtr leads that f the rtr and prduces an electrmechanical trque. Here as well, the rtr and statr fluxes rtate in synchrnism with each ther and that trque is related t the relative displacement between them. Hwever, unlike a synchrnus machine, the rtr f an inductin machine des nt itself rtate synchrnusly; it is the "slipping" f the rtr with respect t the synchrnus armature flux that gives rise t the induced rtr currents and hence the trque. Inductin mtrs perate at speeds less than the synchrnus mechanical speed. A typical speed-trque characteristic fr an inductin mtr is shwn

14 The armature winding f a dc generatr is n the rtr with current cnducted frm it by means f carbn brushes. The field winding is n the statr and is excited by direct current. A cutaway view f a dc mtr: A very elementary tw-ple dc generatr is shwn: The armature winding, cnsisting f a single cil f N turns, is indicated by a and a in diametrically ppsite pints n the rtr. Elementary dc machine with cmmutatr. The rtr is nrmally turned at a cnstant speed by a surce f mechanical pwer cnnected t the shaft.

15 The air-gap flux distributin usually apprximates a flat-tpped wave, rather than the sine wave fund in ac machines, and is shwn in figure: Rtatin f the cil generates a cil vltage which is a time functin having the same wavefrm as the spatial flux-density distributin. Althugh the ultimate purpse is the generatin f a direct vltage, the vltage induced in an individual armature cil is an alternating vltage, which must therefre be rectified. The utput vltage f an ac machine can be rectified using external semicnductr rectifiers. This is in cntrast t the cnventinal dc machine in which rectificatin is prduced mechanically by means f a cmmutatr.

16 In the cnventinal dc machine the cmmutatr is a cylinder frmed f cpper segments insulated frm each ther by mica r sme ther highly insulating material and munted n, but insulated frm, the rtr shaft. Statinary carbn brushes held against the cmmutatr surface cnnect the winding t the external armature terminals. The cmmutatr and brushes can be seen in the figure: The need fr cmmutatin is the reasn why the armature windings f dc machines are placed n the rtr. Fr the directin f rtatin shwn in the figure, the cmmutatr at all times cnnects the cil side, which is under the suth ple, t the psitive brush and that under the nrth ple t the negative brush. Fr the directin f rtatin shwn in the figure, the cmmutatr at all times cnnects the cil side, which is under the suth ple, t the psitive brush and that under the nrth ple t the negative brush. The cmmutatr prvides full-wave rectificatin, transfrming the vltage wavefrm between brushes and making available a unidirectinal vltage t the external circuit.

17 The effect f direct current in the field winding is t create a magnetic flux distributin which is statinary with respect t the statr. Similarly, the effect f the cmmutatr is such that when direct current flws thrugh the brushes, the armature creates a magnetic flux distributin which is als fixed in space and whse axis, determined by the design f the machine and the psitin f the brushes, is typically perpendicular t the axis f the field flux. Thus, just as in the ac machines discussed previusly, it is the interactin f these tw flux distributins that creates the trque f the dc machine. If the machine is acting as a generatr, this trque ppses rtatin. If it is acting as a mtr, the electrmechanical trque acts in the directin f the rtatin. ROTATING MACHINES MMF OF DISTRIBUTED WINDINGS

18 Mst armatures have distributed windings, which are spread ver a number f slts arund the air-gap periphery, as in the figures. Statr f a 190-MVA three-phase 12- kv 37-r/min hydrelectric generatr. Armature f a dc mtr. The individual cils are intercnnected s that the result is a magnetic field having the same number f ples as the field winding. MMF f Distributes Windings The study f the magnetic fields f distributed windings can be apprached by examining the magnetic field prduced by a winding cnsisting f a single N-turn cil which spans 180 electrical degrees, as shwn A cil which spans 180 electrical degrees is knwn as a full-pitch cil. Fr simplicity, a cncentric cylindrical rtr is shwn. The general nature f the magnetic field prduced by the current in the cil is shwn by the dashed lines.

19 MMF f Distributes Windings Since the permeability f the armature and field irn is much greater than that f air, it is sufficiently accurate here t assume that all the reluctance f the magnetic circuit is in the air gap. Frm symmetry f the structure it is evident that the magnetic field intensity H ag in the air gap at angle θ a under ne ple is the same in magnitude as that at angle (θ a + π) under the ppsite ple, but the fields are in the ppsite directin. Arund any f the clsed paths shwn by the flux lines the mmf is N i. The line integral f H inside the irn is negligibly small, and thus it is reasnable t neglect the mmf drps assciated with prtins f the magnetic circuit inside the irn. MMF f Distributes Windings By symmetry the air-gap fields H ag n ppsite sides f the rtr are equal in magnitude but ppsite in directin. It fllws that the air-gap mmf shuld be similarly distributed; since each flux line crsses the air gap twice, the mmf drp acrss the air gap must be equal t half f the ttal r Ni/2. The figure bellw shws the air gap and winding in develped frm, i.e., laid ut flat.

20 MMF f Distributes Windings The air-gap mmf distributin is shwn by the steplike distributin f amplitude Ni/2. On the assumptin f narrw slt penings, the mmf jumps abruptly by Ni in crssing frm ne side t the ther f a cil. This mmf distributin will be discussed again latter. MMF f Distributes Windings AC Machines AC Machines Furier analysis can shw that the air-gap mmf prduced by a single cil such as the full-pitch cil in the figure, cnsists f a fundamental space-harmnic cmpnent as well as a series f higher-rder harmnic cmpnents. In the design f ac machines, serius effrts are made t distribute the cils making up the windings s as t minimize the higher-rder harmnic cmpnents and t prduce an air-gap mmf wave which cnsists predminantly f the space-fundamental sinusidal cmpnent. It is thus apprpriate here t assume that this has been dne and t fcus ur attentin n the fundamental cmpnent.

21 MMF f Distributes Windings AC Machines The rectangular air-gap mmf wave f the cncentrated tw-ple, full-pitch cil f can be reslved int a Furier series cmprising a fundamental cmpnent and a series f dd harmnics. The fundamental cmpnent Ғ ag1 is 4 Ni Fag1 = csθa π 2 where θa is measured frm the magnetic axis f the statr cil, as shwn by the dashed sinusid. It is a sinusidal space wave f amplitude ( F 1) ag peak 4 Ni = π 2 with its peak aligned with the magnetic axis f the cil. MMF f Distributes Windings AC Machines Nw cnsider a distributed winding, cnsisting f cils distributed in several slts. Fr example, as in figure bellw that shws phase a f the armature winding f a smewhat simplified tw-ple, three-phase ac machine. Phases b and c ccupy the empty slts. The windings f the three phases are identical and are lcated with their magnetic axes 120 degrees apart. Our attentin is in phase a alne, pstpning the discussin f the effects f all three phases.

22 MMF f Distributes Windings AC Machines The winding is arranged in tw layers, each full-pitch cil f N c turns having ne side in the tp f a slt and the ther cil side in the bttm f a slt a ple pitch away. In a practical machine, this tw-layer arrangement simplifies the gemetric prblem f getting the end turns f the individual cils past each ther. MMF f Distributes Windings AC Machines Figure bellw shws ne ple f this winding laid ut flat. With the cils cnnected in series and hence carrying the same current, the mmf wave is a series f steps each f height 2N c i a (equal t the ampere-turns in the slt), where i a is the winding current. Its space-fundamental cmpnent is shwn by the sinusid. It can be seen that the distributed winding prduces a clser apprximatin t a sinusidal mmf wave than the cncentrated cil.

23 MMF f Distributes Windings AC Machines The amplitude f the fundamental-space-harmnic-cmpnent f the mmf wave f a distributed winding is less than the sum f the fundamental cmpnents f the individual cils because the magnetic axes f the individual cils are nt aligned with the resultant. The F ag1 equatin is mdified fr a distributed multiple winding having N ph series turns per phase: F 4 kw N ph ples ia cs θ π ples 2 ag1 = a in which the factr 4/π arises frm the Furier-series analysis f the rectangular mmf wave f a cncentrated full-pitch cil, and the winding factr kw takes int accunt the distributin f the winding. This factr is required because the mmf's prduced by the individual cils f any ne phase grup have different magnetic axes. MMF f Distributes Windings AC Machines When they are cnnected in series t frm the phase winding, their phasr sum is then less than their numerical sum. Fr mst three-phase windings, k w typically falls in the range f 0.85 t The factr k w N ph is the effective series turns per phase fr the fundamental mmf. The peak amplitude f this mmf wave is ( F 1) ag peak 4 kw N ph i ples = π a

24 MMF f Distributes Windings AC Machines Exercise: MMF f Distributes Windings AC Machines Slutin: Cntin.

25 MMF f Distributes Windings AC Machines Slutin (cntinuatin): MMF f Distributes Windings Because f the restrictins impsed by the cmmutatr, the mmf wave f a dc machine armature apprximates a sawtth wavefrm mre nearly than the sine wave f ac machines. Fr example, the figure shws diagrammatically in crss sectin the armature f a tw-ple dc machine. (In practice, in all but the smallest f dc machines, a larger number f cils and slts wuld prbably be used.) The current directins are shwn by dts and crsses.

26 MMF f Distributes Windings The armature winding cil cnnectins are such that the armature winding prduces a magnetic field whse axis is vertical and thus is perpendicular t the axis f the field winding. As the armature rtates, the cil cnnectins t the external circuit are changed by the cmmutatr such that the magnetic field f the armature remains vertical. Thus, the armature flux is always perpendicular t that prduced by the field winding and a cntinuus unidirectinal trque results. MMF f Distributes Windings The first figure shws this winding laid ut flat. The secnd, shws the mmf wave. On the assumptin f narrw slts, it cnsists f a series f steps. The height f each step equals the number f ampere-turns 2Ncic in a slt, where Nc is the number f turns in each cil and ic is the cil current, with a tw-layer winding and full-pitch cils being assumed. The peak value f the mmf wave is alng the magnetic axis f the armature, midway between the field ples. This winding is equivalent t a cil f 12Ncic A.turns distributed arund the armature. On the assumptin f symmetry at each ple, the peak value f the mmf wave at each armature ple is 6Ncic A.turns.

27 MMF f Distributes Windings This mmf wave can be represented by the sawtth wave drawn in figure. Fr a mre realistic winding with a larger number f armature slts per ple, the triangular distributin becmes a clse apprximatin. This mmf wave wuld be prduced by a rectangular distributin f current density at the armature surface, as shwn. MMF f Distributes Windings It is cnvenient t reslve the mmf waves int their Furier series cmpnents: The fundamental cmpnent f the sawtth mmf wave is shwn by the sine wave. Its peak value is: 8 = 0.81 times the height f the sawtth wave. 2 π This fundamental mmf wave is that which wuld be prduced by the fundamental space-harmnic cmpnent f the rectangular current-density distributin. This sinusidally-distributed current sheet is shwn dashed

28 MMF f Distributes Windings It is cnvenient t reslve the mmf waves int their Furier series cmpnents: The fundamental cmpnent f the sawtth mmf wave is shwn by the sine wave. Its peak value is: 8 = 0.81 times the height f the sawtth wave. 2 π This fundamental mmf wave is that which wuld be prduced by the fundamental space-harmnic cmpnent f the rectangular current-density distributin. This sinusidally-distributed current sheet is shwn dashed MMF f Distributes Windings Nte that the air-gap mmf distributin depends n nly the winding arrangement and symmetry f the magnetic structure at each ple. The air-gap flux density, hwever, depends nt nly n the mmf but als n the magnetic bundary cnditins, primarily the length f the air gap, the effect f the slt penings, and the shape f the ple face. The designer takes these effects int accunt by means f detailed analyses.

29 MMF f Distributes Windings DC machines ften have a magnetic structure with mre than tw ples. Fr example, the figure (a) shws schematically a fur-ple dc machine. The field winding prduces alternate nrth-suth-nrth-suth plarity, and the armature cnductrs are distributed in fur belts f slts carrying currents alternately tward and away frm the viewer, as symblized by the crss-hatched areas. This machine is shwn in laid-ut frm in (b). The crrespnding sawtth armaturemmf wave is als shwn. MMF f Distributes Windings On the assumptin f symmetry f the winding and field ples, each successive pair f ples is like every ther pair f ples. Magnetic cnditins in the air gap can then be determined by examining any pair f adjacent ples, that is, 360 electrical degrees. The peak value f the sawtth armature mmf wave can be written in terms f the ttal number f cnductrs in the armature slts as: where C a = ttal number f cnductrs in armature winding m = number f parallel paths thrugh armature winding i a = armature current, A

30 MMF f Distributes Windings This equatin takes int accunt the fact that in sme cases the armature may be wund with multiple current paths in parallel. Fr this reasn it is ften mre cnvenient t think f the armature in terms f the number f cnductrs (each cnductr crrespnding t a single current-carrying path within a slt). Thus i a /m is the current in each cnductr. This equatin cmes directly frm the line integral arund the dtted clsed path in figure (b) which crsses the air gap twice and enclses C a /ples cnductrs, each carrying current i a /m in the same directin. MMF f Distributes Windings In mre cmpact frm: where Na = Ca/(2m) is the number f series armature turns. Frm the Furier series fr the sawtth mmf wave f fig. (b), the peak value f the space fundamental is given by:

31 Prf. Elisete Ternes Pereira, Nizwa, Spring 2010