Chapter 27 SINGLE-PHASE IM DESIGN 27.1 INTRODUCTION By design we mean “dimensioning.” That is, the finding of a suitable geometry and manufacturing data and performance indexes for given specifications. Design, then, means first dimensioning (or synthesis), sizing, then performance assessment (analysis). Finally if the specifications are not met the process is repeated according to an adopted strategy until satisfactory performance is obtained. On top of this, optimization is performed according to one or more objectives functions, as detailed in Chapter 18 in relation to three phase induction machines. Typical specifications (with a case study) are • Rated power P n = 186.5 W (1/4 HP) • Rated voltage V sn = 115 V • Rated frequency f 1n = 60 Hz • Rated power factor cosϕ n = 0.98 lagging service continuous or shortduty • Breakdown p. u. torque 1.3-2.5 • Starting p. u. torque 0.5-3.5 • Starting p. u. current 5-6.5 • Capacitor p. u. maximum voltage 0.6-1.6 The breakdown torque p.u. may go as high as 4.0 for the dual capacitor configuration and special-service motors. Also starting torques above 1.5 p. u. are obtained with a starting capacitor. The split-phase IM is also capable of high starting and breakdown torques in p. u. as during starting both windings are active, at the expense of rather high resistance, both in the rotor and in the auxiliary stator winding. For two (three) speed operation the 2 (3) speed levels in % of ideal synchronous speed have to be specified. They are to be obtained with tapped windings. In such a case it has to be verified that for each speed there is some torque reserve up to the breaking torque of that tapping. Also for multispeed motors the locked rotor torque on low speed has to be less than the load torque at the desired low speed. For a fan load, at 50% as the low speed, the torque is 25 % and thus the locked rotor torque has to be less than 25%. We start with the sizing of the magnetic circuit, move on to the selection of stator windings, continue with rotor slotting and cage sizing. The starting and (or) permanent capacitors are defined. Further on the parameter expressions are given and steady state performance is calculated. When optimization design is performed, objective (penalty) functions are calculated and constraints are © 2002 by CRC Press LLC Author Ion Boldea, S.A.Nasar………… ……… verified. If their demands are not met the whole process is repeated, according to a deterministic or stochastic optimization mathematical method, until sufficient convergence is reached. 27.2 SIZING THE STATOR MAGNETIC CIRCUIT As already discussed in Chapter 14, when dealing with design principles of three phase induction machines, there are basically two design initiation constants, based on past experience • The machine utilisation factor C u in W/m 3 D 0 2 L • The rotor tangential stress f t in N/cm 2 or N/m 2 D 0 is the outer stator diameter and L-the stator stack length. As design optimization methods advance and better materials are produced, C u and f t tend to improve slowly. Also, low service duty allows for improved in C u and f t . However, in general, better efficiency requires larger C u and lower f t . Figure 27.1 [1] presents standard data on C u LDC 2 0u = (27.1) for the three phase small power IMs. Figure 27.1 Machine utilization factor C u =D 0 2 L (cubic inches) for fractional/horsepower three-phase IMs. Concerning the rated tangential stress there is not yet a history of its use but it is known that it increases with the stator interior (bare) diameter D i , with values of around f t = 0.20 N/cm 2 for D i = 30 mm to f t = 1 N/cm 2 for D i = 70 mm or so , and more. © 2002 by CRC Press LLC Author Ion Boldea, S.A.Nasar………… ……… In general, any company could calculate f t (D i ) for the 2, 4, 6, 8 single phase IMs fabricated so far and then produce its own f t database. The rather small range of f t variation may be exploited best in our era of computers for optimization design. The ratio between stator interior (bore) diameter D i and the external diameter D o depends on the number of poles, on D o and on the magnetic (flux densities) and electric (current density) loadings. Figure 27.2. presents standard data [1] from three sources T. C. Lloyd, P. M. Trickey and Reference 2. In Reference 2, the ratio D i /D o is obtained for maximum airgap flux density in the airgap per given stator magnetisation m.m.f. in three phase IMs (D i /D o = 0.58 for 2p 1 = 2, 0.65 for 2p 1 = 4, 0.69 for 2p 1 = 6, 0.72 for 2p 1 = 8). The D i /D o values of Reference 2 are slightly larger than those of P. M. Trickey, as they are obtained from a contemporary optimization design method for 3 phase IMs. In our case study, from Figure 27.1, for 186.5 W (1/4 HP), 2p 1 = 4 poles, we choose 332 ou m105615.3LDC − ⋅== with L/D o = 0.380, D o = 0.137 m, L = 0.053 m. The outer stator punching diameter D o might not be free to choose, as the frames for single phase IMs come into standardized sizes [3]. For 4 poles we choose from fig. 27.2 a kind of average value of the three sets of data D i /D o = 0.60. Consequently the stator bore diameter D i = 0.6 × 0.137 = 82.8 × 10 -3 m. The airgap g = 0.3 mm and thus the rotor external diameter D or = D i -2g = (82.8-2×0.3)×10 -3 = 82.2 × 10 -3 m. The number of slots of stator N s is chosen as for three phase IMs (the rules for most adequate combinations N s and N r established for three phase IMs hold in general also for single phase IMs see chapter 10). Let us consider N s = 36 and N r = 30. The theoretical peak airgap flux density B g = 0.6-0.75 T.Let us consider B g = 0.705 T. Due to saturation it will be somewhat lower (flattened). Consequently, the flux per pole in the main winding Φ m is LBK 2 gdism ⋅τ⋅⋅⋅ π ≈Φ (27.3) The pole pitch, m1057.64102.82 4p2 D 33 1 i −− ×≈×⋅ π =⋅π=τ (27.4) With K dis = 0.9 © 2002 by CRC Press LLC Author Ion Boldea, S.A.Nasar………… ……… Figure 27.2 Interior/outer diameter ratio D i /D o versus D o . Wb10382.1053.01057.64705.09.0 2 33 m −− ×≈⋅×⋅⋅ π =Φ Once we choose the design stator back iron flux density B cs = 1.3-1.7 T, the back iron height h cs may be computed from m1091069.8 053.05.12 10382.1 LB2 h 33 3 cs m cs −− − ×≈×= ×⋅ × = ⋅⋅ Φ = (27.4) The stator slot geometry is shows on Figure 27.3. © 2002 by CRC Press LLC Author Ion Boldea, S.A.Nasar………… ……… h =18 ts h =1 ws W =1.8 os h =16 ss h =9 cs W =8 2s b =2.5 ts W =5 1s h =1 os D =137 0 D = 82.8 i Figure 27.3 Stator slot geometry in mm The number of slots per pole is N s /2p 1 = 36/(2×2) = 9. So the tooth width b ts is LBN p2 b tss 1m ts ⋅⋅ ⋅Φ = (27.5) With the tooth flux density B ts ≅ (0.8-1.0) B cs m1055.2 053.03.136 410382.1 b 3 3 ts − − ×≈ ⋅⋅ ×× = (27.6) This value is close to the lowest limit in terms of punching capabilities. Let us consider h os = 1 × 10 -3 m, w os = 6g = 6 × 0.3 × 10 -3 = 1.8 × 10 -3 m. Now the lower and upper slot width w 1s and w 2s are ()() ()() m1000.5 1055.2 36 1128.82 b N hh2D W 3 3 ts s wsosi s1 − − ×≈ × − ++π =− ++π = (27.7) () () m1000.8105.2 36 92137 b N h2D W 33 ts s cso s2 −− ×≈× − ×−π =− −π = (27.8) The useful slot height is © 2002 by CRC Press LLC Author Ion Boldea, S.A.Nasar………… ……… m100.16 10 2 8.82 29 2 137 2 D hhh 2 D h 3 3 i oswscs o ts − − ×≈ × −−−=−−−−= (27.9) So the “active” stator slot area A s is ()() 266 ss s2s1 s m101041016 2 00.800.5 h 2 WW A −− ×=×× + = + = (27.9) For slots which host both windings or in split phase IMs some slots may be larger than others. 27.3 SIZING THE ROTOR MAGNETIC CIRCUIT The rotor slots for single phase IMs are either round or trapezoidal or in between (Figure 27.4). r r 1 2 h 2 h or W ~1 or W =3.5 2r W =6 1r W ~1 or W ~1 or h =1 or h =10 2 h or r Figure 27.4 Typical rotor slot geometries With 30 rotor slots, the rotor slot pitch τ sr is 3 3 r r sr 106.8 30 4102.82 N D − − ×= ⋅×⋅π = ⋅π =τ (27.10) With a rotor tooth width b tr = 2.6 × 10 -3 m the tooth flux density B tr is T343.1 053.0106.230 410388.1 LbN p2 B 3 3 trr 1 tr = ⋅⋅⋅ ⋅× = ⋅⋅ ⋅Φ = − − (27.11) The rotor slots useful area is in many cases (35–60) % of that of the stator slot 266 r ss r m1042.47 30 36 1010438.0 N NA 38.0A −− ×=×××= ⋅ = (27.12) The trapezoidal rotor slot (Figure 27.4), with © 2002 by CRC Press LLC Author Ion Boldea, S.A.Nasar………… ……… () m106106.26.8bWhas m101h;m101Wh 33 trsrr1 3 1or 3 oror −− −− ×=×−=−τ≈ ×=×== (27.13) Adopting a slot height h 2 = 10 × 10 -3 m we may calculate the rotor slot bottom width W 2r m1048.3106 1010 1040.472 W h A2 W 33 3 3 s1 2 r r2 −− − − ×=×− × ×⋅ =−= (27.14) A “geometrical” verification is now required. () [] () [] m1049.3106.2 30 10101122.82 b N 2hhhD W 33 3 tr r 21ororr r2 −− − ×=×− ×++−π = =− ⋅++−π = (27.15) As (27.14) and (27.15) produce the same value of slot bottom width, the slot height h 2 has been chosen correctly. Otherwise h 2 should have been changed until the two values of W 2r converged. To avoid such an iterative computation (27.14)–(27.15) could be combined into a second order equation with h 2 as the unknown after the elimination of W 2r . A round slot with a diameter d r = W 2r = 6 × 10 -3 m would have produced an area () 26 2 3 r m1026.28106 4 A −− ×=× π = . This would have been too small a value unless copper bars are used instead of aluminum bars. The end ring area A ring is 266 r 1 rring m10114 30 2 sin2 1 1042.47 N p sin2 1 AA −− ×= ⋅π ×= ⋅π ×≈ (27.16) With a radial height b r = 15 × 10 -3 m, the ring axial length m106.7 b A a 3 r ring r − ×== . 27.4 SIZING THE STATOR WINDINGS By now the number of slots in the stator is known N s = 36. As we do have a permanent capacitor motor the auxiliary winding is always operational. It thus seems natural to allocate both windings about same number of slots in fact 20/16. In general single layer windings with concentrated coils are used. (Figure 27.5) © 2002 by CRC Press LLC Author Ion Boldea, S.A.Nasar………… ……… mm m m m m mmmm mmmmm mmmmm a aaa aaaa aaa a aaaa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 Figure 27.5 Stator winding m-main phase; a-auxiliary phase. With identical coils per slot and phase; the winding factors for the two phases (Chapter 4) are 9698.0 36 sin5 36 5sin 2 sinq 2 qsin K s m s m wm = π π × = α α = 9810.0 36 sin4 36 4sin 2 sinq 2 qsin K s a s a wa = π π × = α α = (27.17) There are 5 (q m = 5) slots per pole per phase for the main winding and 4 (q a = 4) for the auxiliary one. In case the number of turns/phase in various slots is not the same, the winding factor can be calculated as shown below. As detailed in chapter 4, two types of sinusoidal windings may be built (Figure 27.6) Z 1 Z 2 Z 3 Z 1 Z 2 Z 3 Z 1 Z 2 Z 3 Z 1 Z 2 Z 3 slot axis tooth axis a a a a r = 1 r = 1 a.) b.) Figure 27.6 Sinusoidal windings a.) with slot axis symmetry; b.) with tooth axis symmetry. When the total number of slots per pole per phase is an odd number, concentrated coils with slot axis symmetry seem adequate. In contrast, for even number of slots/pole/phase, concentrated coils with tooth axis symmetry are recommended. Should we have used such windings for our case with q m = 5 and q a = 4, slot axis symmetry would have applied to the main winding and tooth axis symmetry to the auxiliary winding. A typical coil group is shown on Figure 27.7. © 2002 by CRC Press LLC Author Ion Boldea, S.A.Nasar………… ……… From Figure 27.7 the angle between the axis a-a (Figure 27.6-27.7) and the k th slot, β k ν (for the ν th harmonic), is () ss k N 2 1k N πν −+ν π γ=β ν (27.18) a a β 1 β 2 β 3 a D pole len g th D e i l ec L Z Z Z Z Z Z 1 2 3 1 2 3 Figure 27.7 Typical “overlapping” winding The effective number of conductors per half a pole ' Z ν is [4] ' 1 n 1k kk ' Zp4N cosZZ νν = νν ⋅= β= ∑ (27.19) N ν is the effective number of conductors per phase. The winding factor K w ν is simply ∑ ∑ ∑ = = ν = ν ν β == n 1k k n 1k kk n 1k k ' w Z cosZ Z Z K (27.20) The length of the conductors per half a pole group of coil, l cn , is [3] () ∑ = β− π +−π+≈ n 1k 1k 1 eeckcn p2 DL2LZl (27.21) So the resistance per phase R phase is con cn1 Cophase A lp4 R ρ= (27.22) © 2002 by CRC Press LLC Author Ion Boldea, S.A.Nasar………… ……… A con is the conductor (magnet wire) cross-section area. A con depends on the total number of conductors per phase, slot area and design current density. The problem is that, though the supply current at rated load may be calculated as A36.2 98.01157.0 5.186 cosV P I sn n s = ×× = ϕη = , (27.23) with an assigned value for the rated efficiency, η n , still the rated current for the main and auxiliary currents I m , I a are not known at this stage. I m and I a should be almost 90 0 phase shifted for rated load and ams III += (27.24) Now as the ratio m a a m N N I I a == , for symmetry, we can assume that 2 s m a 1 1 I I + ≈ (27.25) With a the turns ratio in the interval a = 1.0-2, in general () 90.0700.0II sm −⋅= . The number of turns of the main winding N m is to be determined by observing that the e.m.f. in the main winding () sm V98.096.0E ⋅−= and thus with (27.3) n1wmmmm fkN2E Φπ= (27.26) Finally turns313 609698.010382.12 11597.0 N 3 m = ×××π × = − In our case, the main winding has 10 (p 1 q m = 2 × 5 = 10) identical coils. So the number of turns per slot n sm is coil/turns31 25 313 pq N n 1m m sm = × == (27.27) Assuming the turn ratio is a = 1.5, the number of turns per coil in the auxiliary winding is coil/turns57 9810.024 9698.05.1310 kpq kaN n wa1a wmm sa ≈ ×× ×× = ×× ×× = (27.28) © 2002 by CRC Press LLC [...]... is about 60 % higher than the presumed rated current (2.36 A) while the torque is twice the rated torque Ten ≈ 1Nm It seems that when the slip is reduced gradually, perhaps around 4 % (S = 0.04) the current goes down and so does the torque, coming close to the rated value, which corresponds, to the rated power Pn (27.58) On the other hand, if the slip is gradually increased the breakdown torque region... for the three-phase IM (Chapter 12) Design trials may now start to meet all design specifications The complexity of the nonlinear model of the single phase IM makes the task of finding easy ways to meet, say, the starting torque and current, breakdown torque and providing for good efficiency, rather difficult This is where the design optimization techniques come into play However to cut short the computation... value for the turn ratio a, lies in the interval between 1.5 to 2.0 except for reversible motion when a = 1 (identical stator windings) The starting and breakdown torques may be considered proportional to the number of turns of main winding squared The maximum starting torque increases with the turn ratio a The flux densities in various parts of the magnetic circuit are inversely proportional to the number... reached To complete the design the computation of core, stray load and mechanical losses is required Though the computation of losses is traditionally performed as for the three phase IMs, the elliptic travelling field of single phase IM leads to larger losses [5] We will not follow this aspect here in further detail We can now consider the preliminary electromagnetic design finished Thermal model may... 5.636A 1.5 0.463 1.5 The starting current is not large (the presumed rated source current Isn = 2.36 A,(Equation 27.23)) but the starting torque is small The result is typical for the permanent (single) capacitor IM 27.8 STEADY STATE PERFORMANCE AROUND RATED POWER Though the core losses and the stray load losses have not been calculated, the computation of torque and stator currents... ……… As the slots are identical and their useful area is (from 27.9) As = 104 × 10-6 m , the diameters of the magnetic wire used in two windings are 2 dm = 4 104 × 0.4 4 A s K fill ⋅ = 10 −3 ⋅ ≈ 1.3 × 10 −3 m π n sm 3.14 31 da = 4 104 × 0.4 4 A s K fill ⋅ = 10 −3 ⋅ ≈ 1.0 × 10 −3 m 3.14 57 π n sa (27.29) The filling factor was considered rather large Kfill = 0.4 The predicted current density in the two... involved as the magnet field is rather elliptical, in contrast to being circular for three phase IMs The machine utilization factor Cu and the tangential force density (stress) ft tend to be higher and, respectively, smaller than for three phase IMs They also depend on the type of single phase IM split phase, dual-capacitor, permanent capacitor, or split-phase capacitor For new specifications, the tangential... various parts of the magnetic circuit are inversely proportional to the number of turns in the main winding, for given source voltage The breakdown torque is almost inversely proportional to the sum Rsm + Rrm + Xsm + Xrm When changing gradually the number of turns in the main winding, the rated slip varies with W2m The starting torque may be increased, up to a point, in proportion to rotor resistance Rrm... 264 246 261 > 0.826 0.826 0.859 0.828 (+) 0.836 According to most of them the objective function F (x) is augmented with the SUMT, [7] m P(X k , rk ) = F(X k ) + rk ∑ j=1 1 G j (X k ) (27.68) where the penalty factor rk is gradually decreased as the optimization search counter k increases There are many search engines which can change the initial variable vector towards a global optimum for P (Xk, rk)... 2 ⋅ sin 2 30 Note The rotor resistance and leakage reactance calculated above are not affected by the skin effects © 2002 by CRC Press LLC Author Ion Boldea, S.A.Nasar………… ……… 27.6 THE MAGNETIZATION REACTANCE X mm The magnetisation reactance Xmm is affected by magnetic saturation which is dependent on the resultant magnetisation current Im Due to the symmetry of the magnetisation circuit it . corresponds, to the rated power P n (27.58). On the other hand, if the slip is gradually increased the breakdown torque region is reached. To complete the design. I m . Due to the symmetry of the magnetisation circuit it is sufficient to calculate the functional X m (I m ) for the case with current in the main winding