CHAPTER 6 ALTERNATING-CURRENT INDUCTION MOTORS Chapter Contributors Brad Frustaglio Andrew E. Miller Earl F. Richards Chris A. Swenski William H. Yeadon 6.1 6.1 INTRODUCTION* This chapter covers the most common types of ac induction motors, including polyphase and single-phase motors. Section 6.2 develops the theory of operation first for polyphase and single-phase operation, including combined winding opera- tions for split-phase capacitor-start and capacitor-run motors. Next, the section develops the two-phase motor and develops single-phase motor theory using the revolving-field and symmetrical-components approach. Finally, Sec. 6.3 develops the three-phase polyphase motor theory. Then, Sec. 6.4 develops geometry,flux path,and performance calculations for the most common configurations of single-phase motors. Next, the section discusses the shaded-pole motor using the cross-field theory as developed by P. H. Trickey and S. S. L. Chang. Modified calculation procedures are included. Finally, the section fin- ishes the calculations by developing geometry, flux path, and performance calcula- tions for the polyphase motor. 6.1.1 Stator and Coil Assemblies A typical stator and coil assembly is shown in Figs. 6.1, 6.2, and 6.3.The stator cores discussed in Chap. 3 are wound with magnet wire in order to produce the magneto- motive force (mmf) that produces the torque. * Section contributed by William H. Yeadon,Yeadon Engineering Services, PC. 6.1.2 Windings Stator and coil assemblies for induction motors usually have distributed wound cores. The objective here is to produce a sinusoidal air gap mmf. A coil of wire, as shown in Fig. 6.4, produces a square-wave mmf.A distributed winding improves this condition. Next is an example of how sinusoidal distribution is determined.This is followed by tables showing near-sinusoidal winding distributions for some common pole- tooth combinations. 6.2 CHAPTER SIX FIGURE 6.1 Stator and coil assembly diagram. FIGURE 6.2 Stator and coil assembly lead end. Example. For any slot combination, the effective turn and sinusoidal distribu- tion can be determined as follows 6 Poles, 1 Phase, in 36-Slot Stator 1. Full pitch (90°) = slots ÷ poles = 36 ÷ 6 = 6 teeth 2. Winding span = 5 teeth and 3 teeth ALTERNATING-CURRENT INDUCTION MOTORS 6.3 FIGURE 6.3 Stator and coil assembly opposite lead end. FIGURE 6.4 Air gap mmf waves versus winding distribution. 3. Effective turn factor: 5 teeth = 90°=75°; sin 75°=0.9659 3 teeth = 90°=45°; sin 45°=0.7071 Sum = 1.6730 4. Percentage of total pole turns per coil = effective turn factor per coil ÷ sum × 100 5 teeth ==0.5773 × 100 = 57.73% 3 teeth ==0.4226 × 100 = 42.26% 5. Winding factor K w (sometimes called winding pitch factor K p ) = Α percentage of total turns per coil × effective turn factor per coil 5 teeth = 0.5576 = 0.5773 × 0.9659 3 teeth = 0.2988 = 0.4226 × 0.7071 Sum = 0.8564 K w /pole = 0.8564 Tables 6.1 and 6.2 show sinusoidal winding distributions for even and odd slot combinations, respectively. Polyphase motors can be wound in a lap winding configuration (Fig.6.5) or a con- centrically distributed winding configuration (Fig. 6.6). Single-phase motors are typ- ically wound and inserted in the manner shown in Fig. 6.7. Figure 6.8 shows a typical 0.7071 ᎏ 1.6730 0.9659 ᎏ 1.6730 3 ᎏ 6 5 ᎏ 6 6.4 CHAPTER SIX FIGURE 6.5 Lap-wound polyphase configuration. wound coil ready for insertion. Figure 6.9 shows the stator core assembly ready for coil insertion. Figure 6.10 shows the partially inserted stator and coil assembly. 6.2 THEORY OF SINGLE-PHASE INDUCTION-MOTOR OPERATION* One of the most important concepts in the analysis and design of all electrical rotat- ing machinery, whether it be a generator or motor, is to obtain a circuit model for performance calculations. Single-phase induction motors are no exception. An equally important concept, which is necessary for induction or synchronous motor rotation, is the existence of a revolving magnetic field in the air gap of the motor.Without this there can be no torque development for producing rotation. Ide- ally, a constant-magnitude rotating mmf is preferred. This latter concept of a rotat- ALTERNATING-CURRENT INDUCTION MOTORS 6.5 TABLE 6.1 Sinusoidal Winding Distribution—Even Tooth Spans Slots per pole 2 4 6 8 10 12 14 16 18 K w 18 6.3 9.0 11.6 13.8 15.7 17.0 17.8 9.0 0.808 9.6 12.4 14.7 16.7 18.1 18.9 9.6 0.835 13.7 16.4 18.4 20.0 20.9 10.6 0.873 18.9 21.3 23.2 24.3 12.3 0.909 26.3 28.6 29.9 15.2 0.944 16 7.9 11.3 14.4 17.2 18.9 20.0 10.3 0.812 12.4 15.7 18.5 20.5 21.8 11.1 0.848 17.9 21.1 23.4 24.9 12.7 0.889 25.7 28.5 30.3 15.5 0.928 38.4 40.8 20.8 0.963 12 6.8 13.2 18.6 22.8 25.4 13.2 0.789 14.1 20.0 24.5 27.3 14.1 0.829 23.3 28.5 31.8 16.4 0.883 37.2 41.4 21.4 0.936 65.9 34.1 0.977 9 12.1 22.7 30.6 34.6 0.795 25.7 34.8 39.5 0.855 47.8 52.2 0.929 8 15.3 28.0 36.8 19.9 0.795 33.1 43.4 23.5 0.870 64.8 35.2 0.950 6 26.8 46.4 26.8 0.804 63.4 36.6 0.914 4 60.8 39.2 0.822 1.000 * Sections 6.2 and 6.3 contributed by Earl F. Richards. Tooth span ing magnetic field is important in the understanding not only of single-phase induc- tion motors but of all rotating ac electromechanical devices. In all multiphase symmetrically balanced motor windings, the existence of a rotating magnetic field can be mathematically demonstrated and verified rather eas- ily. The case of the single-phase motor is a bit more difficult to show, but it can be derived mathematically. Let us explore this rotating revolving field concept by beginning with a two- phase symmetrically balanced induction motor operating on a two-phase power sys- tem. This is a motor having at least two sinusoidally (or as near as possible) distributed stator windings, each of the windings being displaced 90° electrical apart in the stator slots. This means a four-pole motor would have four stator windings located 90° electrical apart, which is equivalent to 45° mechanical or spatial. 6.2.1 Two-Phase Operation One might wonder, “why start with a two-phase motor to explain the single-phase motor?” However, a single-phase motor is designed to be “fooled” into acting like an unbalanced two-phase motor on starting, even though it is operating on a single- phase source when running and starting windings exist on the stator. 6.6 CHAPTER SIX TABLE 6.2 Sinusoidal Winding Distribution—Odd Tooth Spans Slots per pole 3 5 6 9 11 13 15 17 19 K w 18 4.6 7.5 10.2 12.5 14.5 16.0 17.1 17.6 0.794 7.8 10.6 13.2 15.2 16.8 17.9 18.5 0.820 11.5 14.2 16.5 18.2 19.5 20.1 0.854 16.1 18.6 20.6 22.0 22.7 0.892 22.2 24.6 26.2 27.0 0.927 16 5.8 9.4 12.7 15.4 17.6 19.2 19.9 0.797 10.0 13.4 16.4 18.7 20.4 21.1 0.829 14.9 18.2 20.8 22.6 23.5 0.868 21.4 24.5 26.5 27.6 0.910 31.1 33.8 35.1 0.946 12 10.3 16.5 21.4 25.0 26.8 0.809 18.3 24.0 27.8 29.9 0.854 29.3 34.1 36.6 0.910 48.2 51.8 0.969 9 18.5 28.3 34.7 18.5 0.821 34.7 42.6 22.7 0.893 65.3 34.7 0.961 8 27.6 33.2 39.2 0.815 45.8 54.2 0.813 6 42.3 57.7 0.856 100.0 0.965 4 100.0 0.923 Tooth span ALTERNATING-CURRENT INDUCTION MOTORS 6.7 FIGURE 6.6 Concentric-wound polyphase configuration. FIGURE 6.7 Concentric winding for a single- phase four-pole motor with two coils per pole. Solid lines indicate main winding coil sets; dashed lines indicate auxiliary winding coil sets. Consider Figure 6.11, showing a conceptual diagram of a balanced two-pole two- phase motor having sinusoidally distributed windings a and b located 90° electrical apart on the stator.The dashed lines indicate a sinusoidal winding distribution. Directions of the magnetic axes of both the a and b windings for the arbitrarily assumed positive direction of current i a and i b are shown.We recognize that the indi- vidual mmf waves will always lie on the two stationary axes but certainly will change 6.8 CHAPTER SIX FIGURE 6.8 Typical wound coil ready for insertion. FIGURE 6.9 Stator core assembly ready for coil insertion. in magnitude and direction when sinusoidal currents flow in the windings.Let us call the special position around the inside of the stator φ s , measured from the assumed positive direction of the a winding axis in a counterclockwise (CCW) direction. Assume each winding has an effective number of sinusoidally distributed turns N s . Then the mmfs in the air gaps (two per winding axis) of each winding can be expressed with the assumptions of an infinite-permeance magnetic circuit and the mmf distributed evenly across the two air gaps in a sinusoidal fashion, as follows: mmf a = i a cos φ S (6.1) and mmf b = i b sin φ S (6.2) N S ᎏ 2 N S ᎏ 2 ALTERNATING-CURRENT INDUCTION MOTORS 6.9 FIGURE 6.10 Partially inserted stator/coil assembly. FIGURE 6.11 Conceptual diagram of a balanced two- pole two-phase motor. where i a and i b are two currents supplied from a balanced two-phase source. Hence, i a and i b can be expressed as i a = ͙2 ෆ I cos ω e t (6.3) i b = ͙2 ෆ I sin ω e t (6.4) where I is the RMS value of the currents and ω e = 2πf e is the frequency associated with the source voltage. By using the trigonometric identity cos x cos y = cos (x + y) + cos (x − y) mmf a = ͙2 ෆ I ΄ cos (ω e t +φ S ) + cos (ω e t −φ S ) ΅ (6.5) Here ω e t and φ S are respectively functions of time and displacement about the air gap on the stator. Let us jump the gun here and determine what must prevail to make mmf a be a constant magnitude. Certainly, if (ω e t +φ S ) and (ω e t −φ S ) are constants, we will have fulfilled our objec- tive. So then let us proceed: (ω e t +φ S ) = A and (ω e t −φ S ) = B where both A and B are constants. If we take the derivatives with respect to time of both these equations, we have ω e +=0 (6.6) and ω e −=0 (6.7) What can this mean? Equation (6.7) indicates that if we traverse around the sta- tor CCW at a constant velocity ω e =ω s and Eq. (6.6) indicates that if we traverse around the stator CW at velocity ω e =−ω S , the arguments in Eq. (6.5) will be con- stant. Really, what we have shown is that winding a alone has produced two revolving mmfs of half amplitude, with one revolving CCW and the other CW. Think for a moment in terms of a single running winding of a single-phase motor— are these not identical? We will pursue this later when we discuss single-phase operation. Now let us turn our attention to the b winding. By a similar procedure with use- ful trigonometric identities, we can expand as follows: mmf b = ͙2 ෆ I ΄ cos (ω e t −φ S ) − cos (ω e t +φ S ) ΅ (6.8) By the same analysis followed for mmf a , we see that again we have two mmf waves rotating CCW and CW of half amplitude. What happens when we find the total air gap mmf t with both windings energized simultaneously? 1 ᎏ 2 1 ᎏ 2 N S ᎏ 2 dφ S ᎏ dt dφ S ᎏ dt 1 ᎏ 2 1 ᎏ 2 N S ᎏ 2 1 ᎏ 2 1 ᎏ 2 6.10 CHAPTER SIX [...]... Note that in Fig 6 .13 the stationary axis q is in the same direction as the assumed positive direction of mmfa , whereas the assumed positive direction of the d axis is the negative of the positive direction of the mmfb axis Note that the q–d axis is orthogonal and also note that Ks = (Ks)−1 = (Ks)t ෆ ෆ ෆ (6.46) where t indicates the transpose of Ks , and (Ks)−1 indicates the inverse of Ks ෆ ෆ ෆ Consider... to Single-Phase Motors For applications where unbalanced conditions occur, C L Fortesque devised a process whereby unbalanced systems can be reformed into a system of balanced networks The number of balanced networks required is the same as the number of degrees of freedom which occur in the unbalanced system For example, an unbalanced three-phase power system, which has three degrees of freedom, requires... which we are able to handle In the case of the two-degree network, the two components of each of the CCW (forward) and the CW (backward) are orthogonal to each other If we define the CCW rotation of the fields of both the main and auxiliary windings as progressing from the auxiliary to the main winding, we can then draw these components—for example, in terms of the currents in these windings, Im and... oppositely rotating mmfs The first term of mmft rotates CCW In terms of a single-phase motor, what does this mean? There can be no starting torque with equal excitation current, which implies that the two windings (start and run windings) are identical in electrical characteristics 2 What if the two windings have different electrical characteristics and a different number of turns? Consider the following:... resistance and reactance, respectively, of the rotor referred to the main winding; that is ΂ ΃r Nm r′2 = ᎏ Nr 2 2 ΂ ΃x Nm x′2 = ᎏ Nr 2 2 and Na a= ᎏ Nm Na = effective turns of auxiliary winding Nm = effective turns of main winding Nr = effective turns of rotor winding Writing the voltage equations around each winding gives the following: ALTERNATING-CURRENT INDUCTION MOTORS FIGURE 6.19 6.31 Simplified single-phase... Mbsr sin θr Rm (6.29) NbsNr Mbsbr = Mbrbs = ᎏ cos θr = Mbsr cos θr Rm (6.30) Rm is the reluctance of the air gaps for a sinusoidal distribution of flux in the air gap Nas and Nbs are the effective turns of the stator windings, and Nr is the effective turns of the two rotor windings The self-inductances of the stator and rotor can be expressed with leakage inductances included as follows: N2 as ᎏ Rm... transformation to perform this step The new set of variables will transform both stator and rotor machine parameters to a stationary axis q–d with the q axis aligned with the mmfa axis of the stator This is the simplest of other available transformations which could also be used Consider Fig 6 .13 for the stator transformation The stator transformaFIGURE 6 .13 Stator transformation to q–d tion is given... Lmbs(Ids + I′dr) (6.74) 6.22 CHAPTER SIX Solutions of Eqs (6.73) and (6.74) could be considered for the starting performance of a single-phase motor For example, we could let the a winding and the b winding be the starting winding In normal single-phase operation, the b winding would be open at approximately 70 percent of synchronous speed From the solution of these equations, one could obtain the transient... windings are electrically orthogonal to each other Each winding produces a double revolving field Therefore, we have two forward and two backward fields produced by the auxiliary and running windings Following the previous discussion of operations of a single winding, then both the running and auxiliary winding have a forward and backward equivalent circuit, with one additional change Each of the four... double-revolving field analysis of the single-phase motor and will use this concept.) With two symmetrically balanced windings, 90° electrical apart and operated from a two-phase balance source, a constant-amplitude rotating magnetic field is obtained The direction of rotation is a function of the current direction in the two windings Let us now relate what happens to the rotating mmf in considering two cases . 6.10 shows the partially inserted stator and coil assembly. 6.2 THEORY OF SINGLE-PHASE INDUCTION-MOTOR OPERATION* One of the most important concepts in the analysis and design of all electrical. stator windings, each of the windings being displaced 90° electrical apart in the stator slots. This means a four-pole motor would have four stator windings located 90° electrical apart, which is equivalent. 6 .13 the stationary axis q is in the same direction as the assumed positive direction of mmf a , whereas the assumed positive direction of the d axis is the negative of the positive direction of