Design Variations 115 the distribution factor (McPherson and Laramore, 1990; Nasar, 1987; Liwschitz-Garik and Whipple, 1961) sm(Nspp6se/2) Nspp sin(0 se /2) where 7TN,„ TT N N N l x v s •Ly sppiyph TT N (5.7) IV sm is the slot pitch in electrical radians For Nspp = 1, kd is equal to as expected, and for the case Nspp = 2, Nph = 3, kd equals 0.966 Thus, for this latter case, the magnitude of the back emf is reduced to 96.6 percent of what it would be if the same number of turns occupied just one slot per pole per phase Despite the fact that (5.6) applies only when the back emf is sinusoidal, (5.6) is commonly used to approximate the back emf amplitude reduction for other distributions as well The underlying reason for using this approximation is that it is better to have some approximation and be conservative rather than have none at all Pitch factor When Nspp is an integer, the distance between the sides of a given coil, i.e., the coil pitch r c , is equal to the magnet pole pitch r p as depicted in Fig 5.4a However, when Nspp has a fractional component, as in Fig 5.46, the coil pitch is less than the pole pitch and the winding is said to be chorded or short-pitched In this case the relationship between the coil pitch and the pole pitch is given by the coil-pole fraction rc int{Nspp) acp = - = — (5.8) 'p *spp where int(-) returns the integer part of its argument As a result of this relationship, the peak flux linked to the coil from the magnet is reduced simply because the net coil area exposed to the air gap flux density is reduced The degree of reduction is given by the pitch factor kp, which is the ratio of the peak flux linked when r c < rp to that when t c = TP Because the peak flux linked determines the magnitude of the back emf through the BLv law (3.12), the pitch factor gives the degree of back emf reduction due to chording For the square wave flux density distribution considered in Chap 4, the pitch factor is easily computed with the help of Fig 5.7a When r c = t p , the flux linked to the winding is 4>G = BGLTP, where L is the length into the page, and when r c < TP the flux linked is (F>G = BGL,TC Chapter e The ratio of these gives a pitch factor of j P _ _ "ce _ — 7T _ Tn _ a where 6ce - 7racp is the coil pitch in electrical radians (a) Stator Back Iron G = 2B g L and when t c < TP the flux linked is 1, the air gap inductance and mutual air gap inductance are reduced from what they would be when Nspp = The slot and end turn leakage inductances remain unchanged The degree of air gap inductance reduction is on the order of kj Since the air gap inductance is small with respect to the sum of the slot and end turn leakage inductances, more accurate estimation of the air gap inductance is usually not necessary However, more accurate prediction of these inductance components can be found in Miller (1989) Cogging Torque Reduction Cogging torque is perhaps the most annoying parasitic element in PM motor design because it represents an undesired motor output As a result, techniques to reduce cogging torque play a prominent role in motor design As discussed in Chap 4, cogging torque is due to the interaction between the rotor magnets and the slots and poles of the stator, i.e., the stator saliency From (3.24) and (4.39), cogging torque is given by cog % de (5.12) where g is the air gap flux and R is the air gap reluctance Before considering specific cogging torque reduction techniques, it is important to note that (f)g cannot be reduced since it also produces the desired motor mutual torque More importantly, most techniques employed to reduce cogging torque also reduce the motor back emf and resulting desired mutual torque (Hendershot, 1991) Chapter e Shoes The most straightforward way to reduce cogging torque is to reduce or eliminate the saliency of the stator, thus the reason for considering a slotless stator design In lieu of this choice, decreasing the variation in air gap reluctance by adding shoes to the stator teeth as shown in Fig 5.2c decreases cogging torque As discussed earlier in this chapter, shoes have both advantages as well as disadvantages The primary advantage is that no direct performance decrease occurs The primary disadvantage is increased winding inductance Fractional pitch winding Cogging torque reduction techniques minimize (5.12) in a number of fundamentally different ways As discussed above, a fractional pitch winding reduces the net cogging torque hy making the contribution of dR/dd in (5.12) from each magnet pole out of phase with those of the other magnets In the ideal case, the net cogging torque sums to zero at all positions In reality, however, some residual cogging torque remains Air gap lengthening Using the circular-arc, straight-line flux approximation, it can be shown that making the air gap length larger reduces dR/dd in (5.12), thereby reducing cogging torque To keep the air gap flux 4>g constant, the magnet length must be increased by a like amount to maintain a constant permeance coefficient operating point Therefore, any reduction in cogging torque achieved through air gap lengthening is paid for in increased magnet length and cost and in increased magnet-tomagnet leakage flux Skewing In contrast to the fractional pitch technique, skewing attempts to reduce cogging torque by making dR/dd zero over each magnet face This is accomplished by slanting or skewing the magnet edges with respect to the slot edges as shown in Fig 5.8 for the translational case considered in Chap The total skew is equal to one slot pitch and can be achieved by skewing either the magnets or the slots Both have disadvantages Skewing the magnets increases magnet cost Skewing the slots increases ohmic loss because the increased slot length requires longer wire In addition, a slight decrease in usable slot area results In both cases, skewing reduces and smooths the back emf and adds an additional motor output term Design Variations 119 Skewing can be understood by considering (5.12) and Fig 5.8 As one progresses from the bottom edge of the magnet in the figure to the top edge, each component of R, AR(6) across the pole pitch rp takes on all possible values between the aligned and unaligned extremes Moreover, as the magnet moves with respect to the slots, the components of R change position, but the resulting total R = X AR{6) remains unchanged Therefore, dR/dd is zero and cogging torque is eliminated Once again, in reality, cogging torque is not reduced to zero but can be reduced significantly As stated above, the benefits of skewing not come without penalty The primary penalty of skewing is that it too reduces the total flux linked to the stator windings From Fig 5.8, the misalignment between each magnet and the corresponding stator winding reduces the peak magnet flux linked to the coil As before, this reduction is taken into account by a correction factor, called the skew factor ks For the square wave flux density distribution, the skew factor is K = - IT (5.13) where se is the slot pitch in electrical radians, (5.7) For a sinusoidal flux density distribution, the skew factor is (McPherson and Laramore, 1990; Nasar, 1987; Liwschitz-Garik and Whipple, 1961) sin(flse/2) (5.14) 6J2 Of these skew factors, (5.13) gives a greater reduction for a given slot pitch However, both equations show that the performance reduction is minimized by increasing the number of slots This occurs simply because increasing the number of slots reduces the slot pitch, which reduces the amount of skew required ko AR(6) magnet with skew magnet without skew Figure 5.8 Geometry for skew factor computations Chapter e A secondary and often neglected penalty of skewing is that it adds another component to the mutual torque, commonly a normal force (de Jong, 1989) According to the Lorentz force equation (3.25) and (4.1), the force or torque generated by the interaction between a magnetic field and a current-carrying conductor is perpendicular to the plane formed by the magnetic field and current as shown in Fig 3.8 When magnets or slots are skewed, the force generated has two components, one in the desired direction and one perpendicular to the desired direction In a radial flux motor as considered in this chapter, the additional force component is in the axial direction That is, as the rotor rotates it tries to advance like a screw through the stator This additional force component adds a small thrust load to the rotor bearings Magnet shaping Though not apparent from (5.12), magnet shape and magnet-to-magnet leakage flux has a significant effect on cogging torque (Prina, 1990; Li and Slemon, 1988; Sebastian, Slemon, and Rahman, 1986; Slemon, 1991) The rate of change in air gap flux density at the magnet edges as one moves from one magnet pole to the next contributes to cogging torque Generally, the faster the rate of change in flux density the greater the potential for increased cogging torque This rate of change and the resulting cogging torque can be reduced by making the magnets narrower in width, i.e., decreasing r m , or by decreasing the magnet length l m as one approaches the magnet edges In either case, the desired mutual torque decreases because less magnet flux is available to couple to the stator windings Detailed analysis of this approach to cogging torque reduction requires rigorous and careful finite element analysis modeling, which is beyond the scope of this text (Prina, 1990; Li and Slemon, 1988) However, Li and Slemon (1988) provide an approximate expression for the optimal magnet fraction AM = TJTP when the slot fraction is AS = WS/TS = 0.5, = n + 0.14 N N„ sppJ-y ph = n + 0.14 „ ~N Nm Air gap length, m Magnet length, m Outside radius, m Inside radius, m Lamination stacking factor Steel core loss density vs flux density and frequency Lamination stacking factor and steel mass density Conductor resistivity and temperature coefficient Conductor packing factor Magnet spacer width, m Magnet remanence, T Magnet recoil permeability Maximum steel flux density, T Slot opening, m Shoe depth fraction (dv + d2)/wtb Lap or wave, single- or double-layer, or other according to function The axial motor length is not specified in this topology, since it does not directly affect the torque produced As a result, the slot depth is not constrained but can be determined by specifying the conductor current density Also, only two radii are specified since flux flow is in the axial direction Geometric parameters From the fixed parameters in Table 6.3 and the topology shown in Figs 6.3 and 6.7, it is possible to derive several important geometric parameters The magnet pole pitches at the inner and outer radii are ^ I n^a Tpo — where 6P = 2v/Nm pitches are tb Figure 6.7 Slot geometry for the dual axial flux motor topology where the coil-pole fraction is given by (5.8) The slot pitches at the inner and outer radii are (6 53) T - P A 'so where 9S = 2TT/NS is the angular slot pitch The slot cross-sectional area available for conductors is rectangular in this case and can be simply expressed as As = wsbd3 (6.54) where wsb is the slot bottom width The unknown parameters in this design are the slot dimensions The width of the stator teeth varies with radius After the tooth width at the slot bottom and back iron width are determined by the magnetic circuit solution, the slot depth can be determined by constraining the conductor current density to be exactly J m a x Magnetic parameters The magnetic parameters to be found are the stator back iron thickness and the stator tooth width Since the magnet produces constant flux density over its surface, the total flux crossing the air gap increases linearly with radius due to increasing magnet width Thus the amount of flux to be supported by the stator back iron increases with radius ... Trapezoidal Motors In practice there are two common forms of brushless PM motors: motors having a sinusoidal back emf, which are commonly referred to as ac synchronous motors, and trapezoidal back emf motors,... commonly called brushless dc motors Of these, the ac synchronous motor has been around the longest, especially with wound field excitation The brushless dc motor evolved from the brush dc motor as power... the reason for calling it a trapezoidal back emf motor The ac synchronous motor differs significantly from the brushless dc motor An ac synchronous motor has sinusoidally distributed windings, where