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Induction Motors and Their Control

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Induction Motors and Their Control CHAPTER 6 INDUCTION MOTORS AND THEIR CONTROL 6.1 INTRODUCTION An induction machine (Fig. 6.1) is one in which alternating currents are supplied directly to stator windings and by transformer action (induction) to the rotor. The flow of power from stator to rotor is associated with a change of frequency, and the output is the mechanical power transmitted to the load connected to the motor shaft. The induction motor is the most widely used motor in industrial and commer­ cial utilization of electric energy. Reasons for the popularity of induction motors include simplicity, reliability, and low cost, combined with reasonable overload capacity, minimal service requirements, and good efficiency. The rotor of an induc­ tion motor may be one of two types. In the wound-rotor motor, distributed windings are employed with terminals connected to insulated slip rings mounted on the motor shaft. The second type is called the squirrel-cage rotor, where the windings are sim­ ply conducting bars embedded in the rotor and short-circuited at each end by con­ ducting end rings. The rotor terminals are thus inaccessible in a squirrel-cage construction, whereas for the wound-rotor construction the rotor terminals are made available through carbon brushes bearing on the slip rings. The stator of a three-phase induction motor carries three sets of windings that are displaced by 120 deg in space to constitute a three-phase winding set. The applica­ tion of a three-phase voltage to the stator winding results in the appearance of a rotat­ ing magnetic field, as we discover in the next section. It is clear that induced currents in the rotor (due to the rotating field) interact with the field to produce a torque that rotates the rotor in the direction of the rotating field. 323 324 INDUCTION MOTORS AND THEIR CONTROL Figure 6.1 Cutaway of a three-phase cage-type induction motor. 6.2 MMF WAVES AND THE ROTATING MAGNETIC FIELD Consider the N-tum coil that spans 180 electrical degrees shown in Fig. 6.2. This coil is referred to as a full-pitch coil (as opposed to fractional-pitch coils spanning less than 180 electrical degrees) and is embedded in the stator slots as shown. We assume that the permeability of the stator and rotor iron paths is much greater than that ofthe air, and therefore that all the reluctance ofthe magnetic circuit is in the two air gaps. The magnetic-field intensity H in the air gap at angle £) is the same in magnitude as that at 8 +n due to symmetry, but the fields are in the opposite direction. The MMF around any closed path in the vicinity of the coil side is Ni. The MMF is uniformly distributed around the sides, and hence we can conclude that the MMF is constant at Ni/2 between the sides (0 = -n/2 to +n/2) and abruptly changes to -Ni/2, as shown in Fig. 6.2. In part (a) of the figure the physical arrangement is shown, and in part (b) the rotor and stator are laid flat so that the corresponding waveforms can be shown in part (c) in relation to the physical position on the periphery. The MMF waveform of the concentrated full-pitch coil is rectangular, as shown in Fig. 6.2(c), and one needs to resolve it into a sum of sinusoidals. There are two reasons for this. The first is that electrical engineers prefer to use sinusoids due to the wealth of available techniques that enable convenient analysis of such waveforms. The second is that in the design of ac machines, the windings are distributed such that a close approximation to a sinusoid is obtained. In resolving a periodic wave­ form into a sum of sinusoids, one uses the celebrated Fourier series, which for our present purposes states that the rectangular waveform can be written Fo = -4Ni [cos 8 + -I cos 38 + ...] 2n 3 (6.1 ) 4Ni F) = --cos f) 1C2 MMF WAVES AND THE ROTATING MAGNETIC FIELD 325 Magnetic axis of coil N-turn coil N (A) Magnetic axis of coil Rotor Ni/2 r----..."",.-t--......----. -Tr/2 8 -Ni/2 (8) (C) Figure 6.2 MMF of a concentrated full-pitch coil: (a) physical arrangement; (b) rotor and stator laid flat; (c) MMF waves. If the remaining harmonics are negligible, we see that the MMF of the coil is approximated by a sinusoidal space waveform. 6.2.1 Distributed Windings Windings that are spread over a number of slots around the air gap are referred to as distributed windings and are shown in Fig. 6.3. Phase a is shown in detail in the figure, and we assume a two-pole three-phase ac configuration. The empty slots are occupied by phases band c. We deal with phase a only, whose winding is arranged in two layers such that each coil of nc turns has one side in the top of a slot and the other coil side in the bottom of a slot pole pitch (180 electrical degrees) away. The coil current is i., The MMF wave is a series of steps each of height 2ncic equal to the ampere con­ ductors in the slot. The fundamental component of the Fourier series expansion of the MMF wave is shown by the sinusoid in the figure. Note that this arrangement provides a closer approximation to a sinusoidal distribution. The fundamental component is given by (6.2) 326 INDUCTION MOTORS AND THEIR CONTROL Axis of phose a (A) Axis of phase a Current sheet • -.___ ---......----i---~-..., /'" . / / I \ ,, ""X»<sz:> (B) Figure 6.3 MMF wave for one phase (full-pitch coil). where K=~k Nph (6.3) 1t w P The factor k.; accounts for the distribution of the winding, Nph is the number of turns per phase; and P is the number of poles. It is assumed that the coils are series connected, and thus ic = iu • MMF WAVES AND THE ROTATING MAGNETIC FIELD 327 The MMF is a standing wave with a sinusoidal spatial distribution around the per­ iphery. Assume that the current ia is sinusoidal such that (6.4) Thus we have (6.5) The preceding relation can be written as (6.6) where Aa(p) is the amplitude of the MMF component wave at time t, Aa(p) = F max cos tot (6.7) with Fmax = KIm (6.8) 6.2.2 Rotating Magnetic Fields An understanding ofthe nature of the magnetic field produced by a polyphase wind­ ing is necessary for the analysis of polyphase ac machines. We consider a two-pole three-phase machine. The windings of the individual phases are displaced by 120 electrical degrees in space. This is shown in Fig. 6.4. The magnetomotive forces developed in the air gap due to currents in the windings will also be displaced 120 electrical degrees in space. Assuming sinusoidal, balanced three-phase operation, the phase currents are displaced by 120 electrical degrees in time. The instantaneous Axis of phase, b \ Figure 6.4 Simplified two-pole three-phase stator winding. 328 INDUCTION MOTORS AND THEIR CONTROL o ~ 2" 3 3 Figure 6.5 Instantaneous three-phase currents. values of the currents are ia = lmcoswt (6.9) l» = Imcos(wt - 120 deg) (6.10) i.. = Imcos(wt - 240 deg) (6.11 ) where 1m is the maximum value of the current and the time origin is arbitrarily taken as the instant when the phase a current is a positive maximum. The phase sequence is assumed to be abc. The instantaneous currents are shown in Fig. 6.5. As a result, we have Aa(p) = Fmax coswt (6.12) Ab(p) = Fmax cos (wt - 120 deg) (6.13) Ac(p) = Fmax cos (WI - 240 deg) (6.14) where Aa(p) is the amplitude of the MMF component wave at time I. At time I, all three phases contribute to the air-gap MMF at a point P (whose spatial angle is lJ). We thus have the resultant MMF given by A p = Aa(p} cos + A a(/1) cos (0 - 120 deg) + Ac(p) cos (0 - 240 deg) (6.15) Using Eqs. (6.12) to (6.14), we have Ap = Fmax[coSOCOSWI + cos(lJ - 120 deg)cos(wt - 120 deg) + cos (0 - 240 deg)cos(wt - 240 deg)] (6.16) Equation (6.16) can be simplified using the following trigonometric identity: 1 cosexcos{J = 2[cos (ex - P) + cos (ex + P)] SLIP 329 As a result, we have I Ap = 2[COS (0 - lOt) + cos (0 + lOt) + cos (0 - lOt) + cos (0 + tot - 240 deg) + cos (0 - lOt) + cos (0 + tot - 480 deg)]Fmax The three cosine terms involving (0 + wt), (0 + tot - 240 deg), and (0 + wt-480 deg) are three equal sinusoidal waves displaced in phase by 120 degrees with a zero sum. Therefore, (6.17) The wave of Eq. (6.17) depends on the spatial position 0 as well as time. The angle lOt provides rotation ofthe entire wave around the air gap at the constant angu­ lar velocity ro. At time t, the wave is sinusoid with its positive peak displaced cot, from the point P (at 0); at a later instant (t2), the wave has its positive peak displaced wt2 from the same point. We thus see that a polyphase winding excited by balanced polyphase currents produces the same effect as a permanent magnet rotating within the stator. 6.3 SLIP When the stator is supplied by a balanced three-phase source, it will produce a mag­ netic field that rotates at synchronous speed as determined by the number of poles and applied frequency Is: 120h ns = -prpm (6.18) The rotor runs at a steady speed n, rpm in the same direction as the rotating stator field.
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