Induction machines and drives

Transcription

1 Chapte 1 Induction machines and dives Table of Contents 1.1 Induction machine basics Machine model and analysis No load and blocked oto tests Induction machine moto dives Poposed execises Histoical notes Feais shot biogaphy... Fo the teache

2 1. Chapte 1: Induction machines and dives 1.1 Induction machine basics The induction machine is also called asynchonous machine. While the latte name in the authos opinion is moe coect (since induction is a phenomenon that plays a fundamental ole both in synchonous and induction machines) it has the dawback to be vey simila to the one of synchonous machines, thus ceating some communication difficulties. As the synchonous machine, the induction machine is a bidiectional electomechanical convete, i.e. a system able to convet electical enegy into mechanical fom (when it opeates as a moto) and vice vesa (when it opeates as a geneato). It is howeve much moe fequently used as a moto than as a geneato, fo the easons that will be discussed late. As egads its constuction the machine stato is substantially identical to the one used fo the synchonous machine, and discussed in chapte 7 (cf. fig. 1.1 a). Fo the induction machine also, the windings ae nomally distibuted along the ai-gap, although fo simplicity s sake in the figues of this chapte they ae epesented concentated in a single tun. Moeove, fo this machine as well, the windings can have a numbe of pole pais p, diffeent fom one. The oto is constituted by a thee-phase system of windings, such as the one epoted in figue 1.1b, but with conductos bas indeed distibuted along the ai-gap and not concentated in single tuns. These thee windings ae connected to each othe, o, as it is commonly said ae shot-cicuited, as pe fig. 1.1 b), efeed to the case in case of windings sta connection. a) a b) b c stato b c a a a c b b c b c a b c shot cicuit Fig The basic stuctue of an induction machine stato (left) and oto (ight). sta cente To be moe accuate, it must be said that the induction machine otos can have two possible constuctions: wound and squiel-cage. The wound oto is constituted by thee coils sta- o delta-connected, whose fee ends ae shot cicuited. A wound oto with a single tun pe phase has the aspect epoted in fig. 1.1 b). Nomally, in a simila way to what discussed fo the synchonous machine, both stato and oto coils ae distibuted along the ai-gap, spanned though seveal slots; and the diffeent tuns of the same coil ae connected in seies using connections at the two machine ends. Induction machines with wound otos have the advantage of giving access, when needed, to the oto windings, mainly fo stating the machine, as discussed in the moe in depth text of sect The moe fequent oto constuction, howeve, is the so called squiel-cage oto, that is depicted in fig. 1..

3 M. Ceaolo - D. Poli: Fundamentals of Electical Engineeing 1.3 A-A A-A view Fig. 1.. The stuctue of a squiel-cage oto: a) coss section A-A showing ion and bas b) pespective view of only the squiel cage. In the ight pat of the figue a view othogonal to the oto axis is epoted, while in the left one a pespective view of just the conductos is shown. The longitudinal conducto bas ae connected to the two font ings. This constuction makes the oto equivalent to a multi-phase system of windings: each phase is constituted by a single ba, one of the two ings is the cente of the sta connection of the bas, the othe is the shot cicuiting connection. The squiel-cage constuction gives no access to the oto cicuits, but is much simple, moe ugged, and cheape than the wound-oto one. Indeed the wound constuction tends to be abandoned, as discussed late. In cases in which fequent stating opeations o even continuous vaiation of speed ae needed (e.g- in the opeation of a lift 1, o an electic ca) a much moe flexible opeation of an induction machine can be obtained by vaying the fequency of feeding the stato, as discussed, in sect. 1.3, although this will equie the addition of a complex powe electonic convete between the machine and the powe supply. To undestand the opeation pinciple of an induction machine, imagine to connect a machine, whose oto is initially standstill, to an extenal thee-phase souce of powe, that, fo the pupose of this easoning, can be consideed as being constituted by an ideal thee-phase set of ideal sinusoidal souces (fig. 1.3). U a + I a U b U c + + I b I c Asynch. machine 3 T, U c I b I c U a I U a b Fig An induction machine, neglecting tansients, absobs a thee-phase set of cuents, because of the symmety of the feeding voltages and of its inne constuction. We analyse the machine unde the hypothesis that the electic pats of the machine ae in steady-state, i.e. all electical tansients can be neglected. Unde these hypotheses we undestand, 1 All lifts ae subject to vey fequent stat-ups. Lifts of vey high buildings, in addition, equie gadual acceleation/deceleation phases, which equie vaiable-speed opeation of the machine.

4 1.4 Chapte 1: Induction machines and dives fom what we know fom the knowledge acquied in chapte 5, that all voltages and cuents in the machine ae sinusoidally-vaying quantities. In paticula we infe that the thee cuents dawn by the machine fom the souce ae sinusoids. We can also conside that the machine has a full thee-phase symmety, in the sense that thee is no eason fo thinking that what happens in phase a should be diffeent fom what happens in phase b o c except fo the time displacement induced by the time displacement of the thee souces. Theefoe, afte this simple qualitative easoning, we can conclude that the thee cuents ae a thee-phase set, as shown in fig. 1.3 as well. This set of cuents, because of the otating field theoem (chapte 7), does ceate a otating field that poduces in any of the oto windings sinusoidally vaying flux linkages that, in tun, induce a thee-phase set of EMFs, by Faaday s law. The EMF ceation of any of the oto conducto can be evaluated using eq. (7.) : v B l v B l (7.) e PQ el el Whee the speed v el is the speed of the consideed oto conducto in a efeence integal with B. Eq,. (7.) is theefoe natually applied to the machine in a fame of efeence integal with B that, indeed, we know being a otating field. The efeence polaity of e PQ is to be set in such a way that v, B, and a vecto going fom Q to P have a ight-hand oientation; theefoe the + making on e must be set using the ight-hand ule, as shown in fig 1.4, fo which B is consideed otating (in a stationay fame of efeence) counteclockwise. F v el Q - e PQ (t) + P abs B B i(t) P + e P Q (t) Fig Induction and foce geneation on a oto of an induction machine (B oto otating counteclockwise, v el oto speed in a fame otating along with B). The stato windings can be ealised using just two poles, o seveal pole-pais. The constuction of multi-pole-pai windings is the same of that of synchonous machines and some details can be thus be found in chapte 7. Moeove, the stato coils ae distibuted along the aigap, so the voltage poduced by N tuns is kn times the voltage of a single tun, whee k is the adimensional facto discussed in sect Theefoe, the total voltage amplitude of a coil containing N distibuted tuns and p pole pais can be expessed using eq. (7.7), i.e.: Eˆ coil kn SB knpωsb The EMFs induced in the oto, because the oto windings ae shot-cicuited, and since we ae neglecting all the electical tansients, ceate a thee-phase set of cuents, that ceate a new otating field, that combines with the one poduced by the stato cuents. Any of the oto conductos is thus subject to a field, due to the combination of stato and oto otating fields, that belongs to the plane othogonal to the conductos itself, and ae tavesed by cuents. Theefoe they ae subject to Loentz s foce (7.5) : F I B l F I B l (7.5) - Q

5 M. Ceaolo - D. Poli: Fundamentals of Electical Engineeing 1.5 in which the diection of F is detemined using the ight-hand ule with vectos I (having the same diection of the conducto, flowing towads the conducto end towads which positive chages flow) and B. The foce has the diection shown in figue 1.4. It tends to move the oto in a diection which tends to oppose the flux vaiation, i.e. in the example given, counteclockwise as B, in a stationay fame. Now that the oto otates what happens into the machine? Indeed the easoning just followed can be epeated: thee still exist thee stato cuents, that ceate a stato otating field that geneates thee EMFs on the oto, which, in tun, ceate thee cuents, and foces on the oto conductos. The field otational speed is the same as ealie, since it depends only on the angula fequency of the supply voltages U and on the numbe of the machine s pole pais p: =/p, The speed of the oto conductos is now, if is the oto adius : vel (Ω Ω) ( p-ω) By effect of this speed EMFs ae induced in the oto that constitute a thee-phase set. The angula fequency of these voltages is elated to the elative movement of oto and stato field: ot p(ω Ω) and theefoe each of the thee induced voltages will have as amplitude: E ˆ coil knot SB. These voltages will induce in the oto coils cuents that constitute a thee-phase balanced set, which in tun geneate a otating field which, in a fame efeence otating with the oto, will otate at the speed: ot /p Ω -Ω. The field poduced by the oto cuents, evaluated in a stationay fame, will then otate at a speed equal to, i.e. the same speed of the stato field. This is an impotant conclusion: RESULT: Rotational speed of the magnetic field poduced by the oto In an induction machine the otating field geneated by the oto s cuents otates at the same speed as the field poduced by the stato s cuents. This speed is called synchonous speed, and is equal to the angula fequency of the souce, divided by the numbe of machine pole pais. As long as the oto speed emains lowe than the synchonous speed, foce will be geneated on the oto conductos, that will globally poduce a positive toque that will tend to acceleate the oto. But when the actual speed appoaches, the eason fo the toque to be geneated,. i.e. a elative motion between oto and otating fields will tend to vanish, and at exactly the synchonous speed, the toque will be zeo, because in a fame of efeence integal with the oto the machine field will be seen to be stationay, and theefoe no EMF is geneated. In the next section a vey simple and effective mathematical model will be pesented that allows to evaluate quantitatively the electical and mechanical quantities of the machine at the vaious speeds. moe pecisely, is distance of the conducto s coss section cente fom the oto s coss section cente.

6 1.6 Chapte 1: Induction machines and dives 1. Machine model and analysis In the pevious section a desciption of the physical phenomena occuing within the induction machine was epoted unde the hypothesis that all tansients ae neglected, thee exists physical symmety of the machine, and the machine itself is connected to a thee-phase balanced system of voltages. Unde these hypotheses also the cuents absobed by the machine is a balanced thee-phase system of cuents, and the behaviou of the machine can be analysed using the single-phase equivalent concept (if the eade does not ecalls about it they ae suggested to etun to the elevant section of chapte 5). Hee this equivalent cicuit is pesented and its usage is discussed; it is not deived fom the electomagnetic equations only to educe the mathematical buden to the eade. Befoe intoducing the single-phase equivalent of an asynchonous machine a vey impotant quantity is to be defined, the so-called slip s of the machine: Ω Ω s Ω When a machine is standstill its speed is zeo, and the slip is unity. When the oto otates at the same speed of the otating fields, i.e. when no cuent is induced in the oto and theefoe no toque is poduced, slip is zeo. In all the othe cases some powe is conveted between the electical cicuit and the mechanical shaft of the machine. Example 1 A six-pole 6-Hz induction moto uns with a slip of 4%. Detemine the synchonous speed n, the oto speed n, the fequency f of oto cuents, the speed of the oto otating field with espect to the oto (n ) and with espect to the gound (n ). Expessing speeds in Revolutions Pe Minute (RPM): 6 f 66 n 1 RPM p 3 n(1 s) n (1. 4) 1=115 RPM f = sf =.46 =.4 Hz 6 f 6. 4 n 48 RPM p 3 n n n RPM n Hence stato and oto fields ae synchonous. f n Ω ad/s p 6 n Ω (1 s )Ω 1. 6 ad/s 6

7 M. Ceaolo - D. Poli: Fundamentals of Electical Engineeing 1.7 Using the slip, a single-phase equivalent of a synchonous machine can be ceated as epoted in fig I s R s X s I X R P mg /3 + U s - X i I i R i R 1-s s stato ai-gap oto Mechanical shaft Fig Single-phase equivalent cicuit of an induction machine. The eade has suely noted a stong similaity to the tansfome cicuit. Indeed the vey mechanism of powe tansfe fom the stato to the oto has a lot of similaity with the powe tansfe fom pimay to seconday windings of a tansfome. The eade is invited to find himself similaities between the two machines, as soon as he eads and studies this chapte. In the figue two cuved-dashed lines ae epoted, to indicate impotant physical tansfomations: the leftmost pat of the cicuit model the stato quantities, then the dashed cuve named ai-gap is cossed, and the cicuit section containing a desciption of what happens in the oto is shown. The ight-most section of the cicuit epesents the mechanical shaft. Let fist analyse the cicuit in tems of powe. The powes shown in the cicuit ae, obviously, always one thid of the ones ciculating in the actual machine. Theefoe the powe enteing at the left-most teminals: P=U s I s cos is one thid of the powe actually enteing the machine though its teminals. Pat of the powe enteing the machine is dissipated in the esistance of the stato s coils, and that powe is P ls 3Rs I s Anothe dissipation occus in the stato and oto ion, because of paasitic cuents and hysteesis. this is simulated by the esistance R i, and the powe coespondingly dissipated constitutes, the so-called ion losses 3 : P li 3Ri Ii then the powe taveses the ai-gap and the powe enteing the oto, the so-called ai-gap powe is P ag =P-P ls -P i Pat of the ai-gap powe is dissipated in the otos coils, and the powe lost thee is: P l 3R' I It must be noted that the esistance R, and the cuent I ae not exactly esistance and cuent of the oto s coils, but they ae is connected to them. It can be shown to be: k N s s R' R I I k N 1 ' (1.1) in which N s and N ae the tuns of stato and oto coils, espectively and k s and k ae the coesponding coil distibution factos, and is the equivalent tuns atio. 3 the subscipt i stands fo ion.

8 1.8 Chapte 1: Induction machines and dives These expessions imply that the powe dissipated in the thee esistos R by effect of the thee cuents I is exactly the powe dissipated in the oto s coils: P l 3R' I' 3R I The ight-most pat of the cicuit models the powe behaviou of the shaft. Indeed the component indicates ad a esisto having as esistance: 1 s R m, eq R' (1.) s is a fictitious esisto, since the electical powe dissipated in it is actually conveted into mechanical fom. The ight-most dashed cuve in the cicuit is the bodeline between the electic domain of the machine and its mechanical domain. If the mechanical powe geneated in the machine is called P mg, it can be thus witten: P mg 3Rm, eqi' This mechanical powe is also called developed powe. With the obvious deduction of mechanical losses in the beaings and in the esistance encounteed by the oto by effect of the ai suounding it, we obtain the useful (net) mechanical powe, available at the machine flange. The powe flows in the machine ae summaised in figue 1.6. Input electical powe flange (useful) mechanical powe stato coppe losses ion losses oto coppe losses mechanical losses Fig A qualitative chat illustating diffeent types of machine losses. Example A 5-Hz thee-phase induction moto has a ated voltage of 4 V (phase-to-phase). When opeating at full load, it develops its ated mechanical powe of 18 kw at 75 RPM, absobing a line-cuent of 35 A and an electical powe of 1 kw fom the electic gid. Calculate: a) the synchonous speed n b) the slip s c) the powe facto cos d) the toque T e) the efficiency With p=1, the synchonous speed n would be 6f=3 RPM. Since n=75 RPM, 3/n=4.5, then p=4.

9 M. Ceaolo - D. Poli: Fundamentals of Electical Engineeing f 65 n 75 RPM p 4 n n s = =.6 n 75 P 1 cos el UI n Ω ad/s 6 Pmech 18 T = Nm Ω Pmech % P 1 el Example 3 In a fou-pole 5-Hz induction moto, the powe cossing the ai gap and the developed mechanical powe ae espectively kw and.8 kw. Calculate the coppe oto losses and the slip. If the otational losses ae 45 W, detemine the net output toque. The oto losses ae the diffeence between the powe cossing the ai gap and the goss deliveed powe. Since in unning conditions ion oto losses ae negligible (due to the low fequency of oto voltages), such a diffeence coesponds to the coppe oto losses: P cu- =P ag -P d =-.8=1. kw Since P cu- =sp ag, s=1./=5.45% The net mechanical powe can be calculated subtacting otational losses fom P d : P mech =.8-.45=.35 kw f (1 s ) ad/s p Pmech 35 T = 137Nm Example 4 A 38-V thee-phase wye-connected induction moto has a.7+j1.4 pe phase stato impedance. The oto impedance efeed to stato is.6+j1.5 pe phase. The magnetising eactance X i is 4, the tansvese esistance R i is 15. At 4% slip calculate the input cuent, the powe facto, the powe cossing the ai-gap, the mechanical powe and the efficiency.

10 1.1 Chapte 1: Induction machines and dives I s R s X s I X R + U s - Zs X i I i R i Z Rl R 1-s s stato oto Z s =.7+j1.4 Z =.6+j1.5 R l =R (1-s)/s=.6(1-.4)/.4=14.4 Z -tot = Z +R l = 15+j1.5 (also R /s+jx ) The total impedance seen by the stato is: Z tot =Z s +jx i R i Z -tot = 1.7+j6.43 = Hence: Powe Facto=cos(7.66 )=.886 U s =38/3=19.4 V I s =U s /Z tot =14.-j7.351 I =(U s -I s Z s )/Z -tot = 13.6-j.7 = P mg =3R l I = =759 W P ag = P mg /(1-s)=796 W P el =3Re(U s I s * )=93 W = P mg /P el =759/93=8.% Altenative method: P ag =3Re((U s -I s Z s )I s * )=796 W P mg =(1-s)P ag =759 W Let us now apidly conside the meaning of the eactances pesent in the machine equivalent cicuit: eactance X s =L s, whee L s is the popotionality coefficient between the stato cuent and the stato leakage flux i.e. the quote of the flux linkage ceated by the stato cuents that does not coss the ai gap, and theefoe does not inteact with the oto eactance X = L, whee L is the popotionality coefficient between the oto cuent and the oto leakage flux i.e. the quote of the flux linkage ceated by the oto cuents that does not coss the ai gap, and theefoe does not inteact with the stato eactance X i =L i whee Li is the popotionality coefficient between the stato cuents and the flux linkage that cosses the ai-gap. The equivalent cicuit has longitudinal components R s, X s, R, X, and tansvesal components X i, R i. In the nomal opeation of the machine, all the longitudinal components ae much smalle than the tansvesal one. Only when s is nea to one the situation is diffeent, since the load esisto R m,eq is null. In this situation, that occus at the stat-up of the machine, vey lage cuents ae dawn fom the mains and flow though the machine, a situation that is simila

11 M. Ceaolo - D. Poli: Fundamentals of Electical Engineeing 1.11 to shot-cicuit conditions in powe systems. Fo this eason, the longitudinal impedance (R s +R )+j(x s +X ) is also called shot cicuit impedance (Z sc ). In fact, dividing by Z sc the supply voltage, we obtain the shot-cicuit cuent, which coesponds both to the stating cuent and to the cuent absobed by the machine when it is kept blocked and is fed fom the mains. As discussed below, this test is used by enginees to evaluate the numeical values of the equivalent cicuit paametes, obviously at educed voltage to avoid damages due to the high cuents that would othewise ciculate. This gives us a fist indication that the stat-up cuents of this kind of machines ae much lage than the cuents occuing in steady-state, and theefoe eithe fequent stat-ups must be avoided, o the machine must be dimensioned to withstand these lage cuents. A typical application in which the machine opeates with fequent stat-ups is in lifts. Since the single phase equivalent cicuit contains only eactos and esistos, it is obvious that in all its opeating conditions it absobs active and eactive powe, and theefoe its powe facto is always lagging. Indeed the statement that the machine always absobs active powe is to be coected. Since the load esistance is a fictitious one and, fo negative slips, it has a negative value. Unde these conditions, i.e. when the machine oto otates at a speed lage than, the esisto R m,eq actually delives powe, and the machine opeates as a geneato. Even when the machine geneates active powe, howeve, is still absobs eactive powe, since the eactive components of the equivalent cicuit emain positive even at speeds lage than. The single-phase equivalent enables to detemine impotant paametes as a function of the otational speed of the machine oto. Hee it is used to detemine the cuent absobed fom the supply netwok and the mechanical toque geneated. Hee, fo simplicity s sake, this is done in an appoximated way, i.e. neglecting the effects of the tansvesal components R i and X i. This will intoduce some eo, but the esults that will be obtained ae qualitatively coect. The simplified cicuit can be expessed in one of the two foms epoted in figue 1.7, in which is, obviously, X=X s +X. Note that if the two esistances R and R m,eq ae summed up, the esulting esisto will absob a powe that is the sum of the geneated mechanical powe and the oto s coppe losses, that can be imagined as the powe cossing the ai-gap and enteing the oto. This powe is theefoe called P ag. I s R s X R I s R s X + U s - + U 1 - R 1-s s + U s - + U 1 - R s P mg /3 Fig Simplified cicuits of an induction machine. We want now to deive the cuent absobed and toque poduced unde the condition U s =const. Indeed, as a futhe appoximation, we conside hee constant the voltage U 1, that can be measued downsteam R s that does not diffe significantly fom U s. Using the ight cicuit in figue 1.7, the following can be witten: I U R' P ag / P ag 3 I 3 3 X R' / s s s X R' / ' s s X R R' U U R' s

12 1.1 Chapte 1: Induction machines and dives and: T mg 3 U1 R' s Pag / Pmg / (1.3) s X R' in which the symbol T mg indicates the mechanical, geneated toque. The useful toque T mu will be lowe than this by effect of mechanical losses. If we imagine the oto speed to go fom zeo to, the coesponding slip will go fom 1 to ; the denominato of the faction descibing I will steadily gow, and theefoe the cuent I will decease monotonically. To analyse the shape of T mg () conside fist that two impotant points of the cuve ae those coesponding to s= and s=1. It is obviously: T mg ( s ) T mg T mg 3 ( ) U1 R' X R ' 3 U R' 1 It is also of inteest to evaluate whethe between these two points thee is a peak. This is done equating to zeo the fist deivative of T mg (s): Tmg 3U 1 sˆ R' / X Tˆ mg Tmg (ˆ) s (1.4) s X It can be easily veified that it is always Tˆ mg T mg and theefoe the cuve has a maximum fo s sˆ. The coesponding angula speed will be called ˆ. Note that while ŝ depends on the oto s esistance, Tˆ mg does not. Compaing the expession of I and equation (1.3), it is possible to obseve that st 3R' = = constant (1.5) I fo any woking condition. This obsevation is vey useful to coelate T (s=), T max and T full-load (T ated ) to thei coespondent cuents. Z Example 5 An induction moto has a slip of 4% at full load. The stating cuent is 6 times the fullload cuent. Calculate the atio of the stating toque to the full-load toque. Fom Eq (1.5): Tstating Istating s full-load Tfull-load Ifull-load sstating 1 Example 6 Fo the moto of Example 4, calculate the maximum mechanical powe and the coespondent slip ŝ and toque T, the fequency being 5 Hz and the pais of poles in numbe of. X= X s +X =1.4+1,5=.9 s =R /X=.6/.9=.7 =f/p=157.1 ad/s 3U T 158, 5 Nm Ω X

13 M. Ceaolo - D. Poli: Fundamentals of Electical Engineeing 1.13 P mg = (1-s) T =1975 W Please note that the equation used to calculate T does not conside R s. Moe in geneal, it is possible to demonstate that: 3 U1 R' s T Ω s X ( srs R' ) R' 3U1 ŝ, Tˆ R X Ω ( R R X ) s s s Typical shapes of the absobed cuent (RMS value) and mechanical toque geneated, as a function of the otational speed, ae epoted in fig These cuves wee deived using ealistic numeical paametes of a 5 kw squiel-cage machine. This machine has a peak toque that is.35 times the stating toque The stating toque can be highe than this value, but this is to be obtained using a lage oto esistance, and this educes the machine efficiency. To have high efficiency the coppe losses, and the oto esistance, must be small, but this educes the stating toque at equal peak toque. The toque, as expected, becomes zeo when the synchonism speed is eached, then it eveses, and the machine opeates as a geneato. Obviously, fo the geneato egion to be actually eached it is necessay that at the machine shaft a mechanical moto is attached, i.e. a device able to supply mechanical powe. Nomally, howeve, as aleady noted, induction machines ae opeated as motos. cuent toque T 1 P Q moto opeation geneato opeation ˆ Fig A typical toque vs. angula speed induction machine shape of an induction machine. The cuves epoted in figue 1.8 clealy show that the nomal opeation zone of the machine is between the peak toque (eithe positive o negative) and the synchonous speed. Fo example the toque T 1 can be deliveed at the speed coesponding to point P and Q; but point P, while coesponding to a much lowe mechanical powe, implies the absoption of a much highe cuent and losses! In the next section it will be shown that using a moto dive it is possible to delive a given toque at diffeent speeds, while absobing oughly the same cuent fom the supply.

14 1.14 Chapte 1: Induction machines and dives The zone between = and ˆ is nomally efeed to as an unstable egion, while the egion between ˆ and is consideed stable. This is not totally coect since stability depends on the dynamic behaviou of a system, while the toque cuves we ae discussing ae stationay, but tells some tuth: in the moe common cases, indeed, equilibium points between the toque chaacteistic of the machine and of the load ae stable only (in the machine opeation as a moto) between ˆ and. The opeation of an induction machine as a moto with its mechanical load is depicted in figue 1.9, whee both the motive toque T m poduced by the machine and the esistive one absobed by the load T ld (just as an example hee is a fan) ae epoted. The diffeence between the two is the acceleating toque. unst st P m ld Q M 3 T, Fig Taction and load toques of an induction machine with a mechanical load. Just to give a qualitative explanation of stable and unstable egions of the toque cuve of the machine conside the system to opeate at the equilibium point P between the machine toque and a possible load toque T unst. In case a light incease in occus, the machine toque becomes lage than the load, and the speed tends to futhe incease. Also in case of a speed decease the system tends to move fa fom the equilibium point. This does not happen in case of point P and cuve T st, no in case of point Q, of intesection with cuve T ld, that show a stable behaviou. Fom the figue one can even have an idea of a stating-up tansient of the machine, using the mechanical equation: T T J mg ld In which obviously J is the combined moment of inetia of machine and mechanical load. It must howeve be claified that this is valid only as a fist appoximation, since the cuves epoted in figues 1.8 and 1.9 ae computed using the equivalent cicuit that was dawn assuming steady-state opeation of the machine, while duing the tansient, especially the fist pat of it, the actual behaviou is athe diffeent. The shape of the toque of an induction machine has the inconvenience of a stat-up value is smalle, often much smalle, than the peak toque and that duing stating-up the cuents absobed by the machine ae vey high. This can be mitigated by the usage of wound-oto machines and solved, but a lage cost, adopting electic dives instead of simple machines. As has been seen ealie, wound oto machines allow to access the oto windings, to connect them electically to a stationay cicuit, typically fo limited duations. Conside now that as shown in (1.4), the slip at which the machine toque has its maximum is s=r /X= R /X, and is theefoe popotional to the oto esistance, while the coesponding maximum does not depend on this esistance. Wound-oto machines can thus be stated inseting in seies with the oto windings at slow speed additional esistos R 1, R,... that ae pogessively by-passed, as fa as the speed inceases.

15 M. Ceaolo - D. Poli: Fundamentals of Electical Engineeing 1.15 This is done collecting the oto coils cuents though slip-ing/bush coupling, of the same type of those pesent in synchonous machines, using thee ings to get the thee-phase set of cuents. When the system is nea it steady-state speed, all the extenal esistos ae by-passed, and, by means of a special mechanical aangement, the oto windings ae shot.-cicuited, and the bushes lifted fom the ings, to avoid fiction and bush consumption (fig. below). 1 closes closes Supply 1 R 1 R 3 M T, This system is now becoming obsolete, since has seveal disadvantages ove the moe moden solution of the usage of induction moto dives, that ae discussed in the next section: the extenal esistos poduce excess losses, the ing-bush connection is mechanically complicated and costly, and equies maintenance; the change between the diffeent toque cuves is discete and not continuous. When an induction machine is equied to opeate at continuously-vaiable speed, such as in electic tains o cas, the mechanical chaacteistic of the machine fed by constant voltage/constant fequency souce, such as that shown in fig. 1.8 and 1.9, is totally inadequate. In these cases induction machines can be used, and ae today indeed fequently used, using special devices to feed them, that ae able to geneate voltages having desied fequency and amplitude. The machine and the feeding system constitute a subsystem that is called moto dive. Some basic infomation about induction machine-based moto dives is epoted in the following section. 1.3 No-load and blocked-oto tests A method fo detemining the paametes of the equivalent cicuit of an induction machine consists of two tests: the no-load test and the blocked-oto test. In the fome, the ated voltage is applied to the machine, while living the oto fee of otating (no mechanical load is applied). The cuent I absobed by the moto is measued, as well as the coespondent active powe P (absobed by R s and R i ). Since the cuent is small, P is a good estimation of stato ion losses 4 (at low slip, oto losses ae always negligible in an induction machine, since the oto fequency is low). Such losses, which stictly depend on the voltage applied to R i, emain pactically constant fo any loading condition, povided that the machine is supplied at its ated voltage. R U P i, U U P Xi, whee cos Q P tan 3UI The second test is pefomed inceasing the supply voltage until ated cuent is absobed by the moto, the oto being mechanically blocked. The voltage U sc is then measued, as well as the active powe P cc. Since U sc is usually only some pecents of the ated voltage, duing this test I i can be neglected espect to I s (the coe is pooly magnetised); thus the powe absobed by R i is 4 If R s is known, fo example thanks to a pevious DC measue, to estimate the ion losses P can be coected by subtacting 3R s I.

16 1.16 Chapte 1: Induction machines and dives much lowe than the one consumed by R s and R. Fo this eason, P sc is a good estimation of coppe losses at ated cuent; in any othe loading condition: Pcu Psc ( I/Iated ) The test is unable to sepaate R s +jx s fom R +jx, but it allows calculating Z sc : Usc jφ sc Psc Z sc e, whee cossc 3Iated 3UscIated Finally, electical and mechanical powes ae stictly elated: Pel Pmech PFe PCu The eade should note the stict analogy with open-cicuit and shot-cicuit tests of a tansfome. Example 7 A 38-V 8-kW thee-phase 4-poles induction moto has issued the following esults to the no-load and blocked-oto test: I =5%, P =1.% U sc =1%, P sc =7% The stato esistance is.1 (phase) The ated line cuent is 15 A and the stato is delta-connected. Calculate R i, X i and Z sc I =.515=.75 A P =.18=96 W P cos = =.194, tan =5.44 3UI U sc =.1*38=45.6 V (phase-to-phase) P sc =.7*8=56 W Psc cos sc = =.473, sc =61,8 3 U I sc ated The wye-connected equivalent stato esistance is.1/3=.4. The ion losses can be calculated subtacting the no-load stato coppe losses fom P : P Fe = =95.93 W (no-load stato coppe losses ae negligible) R i =U /P Fe =155 Neglecting X s : X i =U /(P tan )=98. Z sc = U sc 3 Iated =1.755 Z sc = Z sc sc = =.89+j1.546 = (R s +R )+j(x s +X ) 1.4 Induction machine moto dives Conside the above deived expessions of the machine cuent and toque:

17 M. Ceaolo - D. Poli: Fundamentals of Electical Engineeing 1.17 is: I U 3 U R' s (1.3) 1 1 T mg Pag / Pmg / X R' / s s X R' Let us evaluate what happens of the expessions when the machine is fed in such a way that U K f f K (1.6) 1 i.e., the machine is fed in such a way that the fequency of supply and voltage U 1 emain always popotional to each othe (emembe that =/p=f/p). Because of its impotance a supply compliant with (1.6) has a name of its own: it is nomally called constant voltage/fequency supply. Using constant voltage/fequency supply and consideing that it is: : s the squaed cuent becomes s U1 K I. (1.7) s X R' L p R' In the latte equality it has been consideed that X is popotional to the angula fequency : s s X X X ( L L ) p L. The toque is: K R' K R' Tmg 3 3 (1.8) s X R' L p R' Equations (1.7) and (1.8) indicate that when the machine is fed accoding to (1.6) the cuent absobed and toque poduced depend only on and not by alone. Since the shape of T mg as a function of does not depend on, it can be deived fom what seen in the pevious section. Evidently if the function T() is the one of pevious figues, epoted again in the left pat of figue 1.1, changing leaves the shape intact, just the cuve is tanslated in such a way that the toque becomes zeo fo the given value of Fig Tanslation of toques feeding the machine with the ule U 1 =K f f. This technique can be used to make the machine opeate so that it opeating point is always in the moe efficient egion of the chaacteistic, i.e. between ˆ and. In this egion, as was seen ealie, the toque is deliveed at the lowest cuent and highest efficiency. As an example, fig shows the stating of the same machine unde two conditions: using a constant teminal voltage and the technique (1.6) Duing the stat-up is continuously changed in such a way that = -=constant, and theefoe the toque geneated accoding to

18 1.18 Chapte 1: Induction machines and dives the theoy developed is constant and lage than the load toque. When the taget speed base is eached, howeve, the supply system woks at that speed and the system eaches its equilibium point. The figue is obtained using a simulation pogam that models the machine with a highe level of detail. To undestand the shown plots it must fist be noted that the time span is eight seconds, theefoe sinusoids opeating at 5 Hz ae seen as a thick band, since they evolve too apidly to be seen well in the shown scale. This is paticulaly tue fo the top-left plot, in which the unifom band is the visual epesentation of a constant-amplitude, constant-fequency 5 Hz sinusoid. Voltages Cuents Toques Constant voltage and fequency supply Supply U 1 =K f f; =const Fig Simulation of the stating of an induction machine eithe fed diectly fom the mains (left) o with vaiable voltage/fequency (ight). The following obsevations can be done: at the beginning of the tansient, in the case of constant-fequency supply, the toque shows a high fequency fluctuation aound a steadily inceasing value. This fluctuation is actual in eal machines, and ae not epoduced by usage of the equivalent cicuit-based model poposed in this chapte, in which all electical tansients ae neglected. In the vaiable-fequency supply the toque fluctuation is educed to a minimum. the vaiable fequency stating allows the machine to be stated-up in the same time, but at a much lowe cuent and educed toque fluctuations; The supply voltages to obtain the cuves epoted in the ight pat of figue 1.11, ae obtained using a contol logic of the type descibed in fig. 1.1 The efeence toque duing the statup T* is fist conveted in a signal (taking into account the inne paametes of the induction machine) but satuated at the value of ˆ to impose opeation of the machine in the moe efficient zone of its toque. Adding the measued value meas of the synchonous speed is detemined, that is conveted into fequency using the pole pais numbe p. Befoe this, the signal is satuated to the efeence value ss, since the machine is intended to be stated up to this value of synchonous speed. Once the fequency is

19 M. Ceaolo - D. Poli: Fundamentals of Electical Engineeing 1.19 known, the voltage U 1 =kf* is known and fom it, using the estimate I* of the cuent computed fom T*, the voltage U* desied at the machine teminals is detemined. The values of the voltage U* and fequency f* to be applied to the machine ae then tansfeed to the supply system that ceates a thee-phase voltage system having that voltage and fequency. T* ˆ + + ss p f* U=kf*+R s I* f* U* Satuation meas ss I* f* U* Supply system 3 M T, meas mechanical load tachomete Fig A possible toque contol logic (used to obtain the plots epoted in fig. 1.11). It is appaent that the vaiable-fequency contol allows the machine to stay stably at any speed, because the cuents ae kept unde contol. The value of detemines the toque and the cuent, the latte being the value coesponding to the moe efficient egion of the toque chaacteistic of the machine, i.e. the one between * and. The opeation of the induction machine unde (1.6) is called also constant flux opeation. This is because the voltage U 1 coesponds in the machine to the EMF poduced by the flux linked with the stato. Fo each phase: U1 j And theefoe U 1 /f= const implies = const. As speed inceases, the voltage to be applied to the machine teminals inceases. When the maximum continuous voltage of the machine is eached (nomally called nominal voltage), this quantity cannot be inceased anymoe, while it may happen that a futhe incease in speed is equied. The speed that coesponds to the nominal voltage is nomally called the base speed of the machine. Beyond the base speed the voltage is kept constant, while is aised, inceasing the fequency of the supply voltages. In these conditions, the cuve of the toque is pogessively educed, being popotional to 1/ : T mg U1R' s U1R' s X R' L p R' 3 3 The full pictue of the machine, showing the constant flux and flux weakening pats, is shown in figue It shows the zone below base, in which the toque chaacteistics tanslate, and. the zone above this value whee they educe in popotion to 1/ A possible shape of the load toque T ld, and a possible locus of the machine opeating points T T J is also epoted. m ld

20 1. Chapte 1: Induction machines and dives T, P T m P T ld base Fig Induction machine dive composed by the constant flux (< base ) and flux weakening (> base ) egions. Using this T m, the mechanical powe is linealy inceasing in the constant flux egion; when base is ovecome, the T m locus shown in figue is such that the powe P is maintained constant and theefoe the actual toque educes in popotion to. Doing so, since the machine toque shape educes in popotion to 1/, the machine opeating point becomes moe and moe nea to the peak toque. Rathe obvious, the zone with the flux eduction cannot be vey lage, because of the coesponding toque eduction. A typical atio of max / base in a nomal induction dive does not ovecome. Some final wods about induction moto dives. The constant voltage/fequency contol allows smooth opeation of the machine, and usage of it at all speeds with good efficiency. Howeve, in fast changing conditions it is not optimal because its theoy comes fom a steady-state model of the machine. Moe advanced contol techniques, nomally called vecto contol, that keep unde contol the inne machine magnetic field ae today used in advanced dives, and show bette behaviou in dynamic conditions. These techniques, howeve, ae well beyond the scope of this book and will not be dealt with. 1.5 Histoical notes Feais shot biogaphy

21 M. Ceaolo - D. Poli: Fundamentals of Electical Engineeing Poposed execises Whee not specifically indicated, electomagnetic quantities ae expessed in RMS values An eight-pole 5-Hz induction moto uns with a slip s=5%. Detemine the synchonous speed n, the oto speed n, the fequency f of oto cuents, the speed of the oto otating field with espect to the oto (n ) and with espect to the gound (n ). Expess all speeds in RPM (Revolutions Pe Minute). 1.. A fou-pole 6-Hz induction moto uns at 175 RPM. Detemine the synchonous speed n, the slip s and the fequency f of oto cuents. 1.3 A six-pole 5-Hz induction moto has a fullload slip of 3%. Detemine: a) the synchonous speed n b) the oto speed n at full load c) the fequency of oto cuents at the instant of stating d) the fequency of oto cuents at full load 1.4 The nameplate speed of a 5-Hz is 7 RPM. If the no-load speed is 745 RPM, estimate the numbe of poles and calculate the synchonous speed and the slip at full load A 5-Hz, thee-phase induction moto has a ated phase-to-phase voltage of 4 V. When opeating at full load, deliveing its ated mechanical powe of 3 HP, the line-cuent is 4 A, the oto speed is 71 RPM and the moto absobs 5.7 kw fom the electic gid. Calculate: a) the synchonous speed n b) the slip s c) the powe facto pf d) the toque T e) the efficiency 1.6. An eight-pole, 5-Hz, thee-phase induction moto has an efficiency of 86% and absobs 48.8 kw fom the electic gid. If the slip is 5%, calculate the shaft toque Induction motos ae often baked evesing the phase sequence of the voltage supplying the moto (plugging). A moto with 6 poles is opeating at 115 RPM while supplied at 6 Hz. Two of the stato supply leads ae suddenly intechanged. Calculate the new slip and the new oto cuent fequency In a fou-pole 5-Hz wound-oto induction moto, stato and oto have the same effective numbes of tuns pe phase. If the moto has V pe phase acoss its stato and the voltage induced in the oto is 6.6 V pe phase, calculate the slip and the moto speed In a two-pole 6-Hz induction moto, the powe cossing the ai gap and the developed powe ae espectively 1.6 kw and.1 kw. Calculate the slip s. If the otational losses ae 4 W, detemine the net output toque A thee-phase six-pole 5-Hz induction moto develops its maximum toque of 61 Nm at 6 RPM. The oto esistance epoted to stato, R, is.5. Calculate the supply voltage (phase to phase) and the toque developed by the moto at 75 RPM An induction moto develops a maximum toque twice the stating toque. Calculate X/R. If the full-load toque equals the stating toque, detemine the full-load slip An induction moto has a wounded oto, having an impedance of.+.6j pe phase. Which is the esistance to be added to each oto phase, in ode to maximise the stating toque? If no esistance is added, at which slip the moto develops the maximum toque? An induction moto has a slip of 4% at full load. The stating cuent is 8 times the full-load cuent. Calculate the atio of the stating toque to the full-load toque. If the moto is stated at educed voltage, so that the stating cuent is only 4 times the full-load cuents, which is the new atio of the stating toque to the full-load toque? Comment this esult An induction moto has a slip of 5% at full load. At ated voltage, the stating cuent is 6 times the full-load cuent. Calculate the atio of the stating toque to the full-load toque. If the moto employs a wye-delta state, which connects the moto phases in wye fo stating and in delta when aleady unning, which is the new atio of the stating toque to the full-load toque? Comment this esult The esistance measued between two stato teminals of an induction machine is.14 ; the stato id delta-connected. The moto delives a

22 1. Chapte 1: Induction machines and dives mechanical powe of 14.7 kw, while supplied at V (phase-to phase), absobing a line cuent of 54 A at.8 powe facto (lagging). The no-load test has issued the following esults: I = A, cos =.1. Calculate: a) the esistance of each stato phase b) the efficiency c) the no-load powe d) the sum of ion and mechanical losses e) the coppe losses at the cuent load condition A thee-phase induction moto has a fullload slip of 4%. The wounded oto is sta connected, with a phase esistance is.1. Calculate the additional oto esistance equied to obtain the full-load toque at stating conditions An eight-pole -V 4-Hz induction moto absobs a line cuent of 6 A with a.8 powe facto (lagging) and a slip of 3.5%. The esistance measued between two stato teminals is.16, the stato being delta-connected. The no-load test has issued the following esults: I =18 A, cos =.1. Calculate: a) the esistance of each stato phase b) the sum of ion and mechanical losses c) the stato coppe losses d) the powe cossing the ai gap e) the powe and toque deliveed f) the efficiency A six-pole induction moto delives kw with an efficiency of 91%, a powe facto of.9 and a slip of 3%. The moto is supplied by a 38 V 5 Hz phase to phase voltage. Calculate the line cuent and the deliveed toque A six-pole -V 5-Hz induction moto absobs kw with a powe facto of.68 and a slip of 3.4%. The esistance measued between two stato teminals is.3, the stato being delta-connected. The ion losses ae 8 W, the mechanical losses 44 W. Calculate: a) the stato coppe losses b) the net mechanical powe and toque c) the efficiency d) the capacity of delta-connected capacitos, equied to compensate the moto to a unit powe facto 1.. A fou-pole induction moto with a wyeconnected stato, having a phase esistance of.4, is supplied at 38 V, 5 Hz. Ion and mechanical losses equal espectively 35 W and 3 W. The moto moves a DC geneato, which delives 7 A at 15 V. The moto absobs 1.5 kw with a.9 powe facto, otating at 1446 RPM. Calculate: a) the oveall efficiency b) the moto efficiency c) the DC geneato efficiency 1.1. A two-pole thee-phase induction moto is supplied by a 38 V 5 Hz voltage. The oto esistance epoted to stato is.15 ; the oveall leakage eactance epoted to stato is 1. Calculate: a) the maximum toque and the coespondent slip and line cuent b) the full-load toque and line cuent, if the fullload slip is 3% c) the stating toque and line cuent 1.. The moto of the pevious execise is applied to a mechanical load equiing a toque of 3+.5 Ω mecc Nm. Using eq. (1.3) and a numeical technique, calculate: a) the slip b) the line cuent c) the oto speed in RPM d) the mechanical powe and toque deliveed to the load 1.3. A 5-Hz 15-kV (phase-to-phase) netwok supplies, by means of a MV/LV tansfome, a 38 V thee-phase 4-pole induction moto, with a delta-connected stato. The tansfome has the following paametes: 15/38 V (phase-to-phase) delta/wye connection 18 kva, 5 Hz I =%, P =.4% U sc =7%, P sc =3% The moto has the following paametes: P mech ated =8 kw I ated =16 A (line) I =6%, P =1.% U sc =14%, P sc =8% R s =.1 (phase) Between the tansfome and the moto thee is an 8-m LV cable, with =.9 /km and x=.15 /km. Thévenin s impedance of the MV netwok can be neglected. Daw the single-phase wye-equivalent of this system. Assuming that the moto is unning with a 3% slip, calculate: a) the moto speed in RPM b) the mechanical powe and toque deliveed to the load

23 M. Ceaolo - D. Poli: Fundamentals of Electical Engineeing 4.3 c) the phase-to-phase voltage at moto mains d) the stato phase cuent of the moto e) the pimay phase cuent of the tansfome

24