Electromagnetic Radial Forces in a Hybrid Eight-Stator-Pole, Six-Rotor-Pole Bearingless Switched-Reluctance Motor Carlos R

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Electromagnetic Radial Forces in a Hybrid Eight-Stator-Pole, Six-Rotor-Pole Bearingless Switched-Reluctance Motor Carlos R. Morrison, Mark W. Siebert, and Eric J. Ho

s Abstract—Analysis and experimental measurement of the A0 Common cross-sectional area between aligned electromagnet force loads on the hybrid rotor in a novel rotor and stator pole, 2×10–4 m2 bearingless switched-reluctance motor (BSRM) have been performed. A BSRM has the combined characteristics of a B Flux density (T) switched-reluctance motor and a magnetic bearing. The BSRM c has an eight-pole stator and a six-pole hybrid rotor, which is Bk Flux density between stator pole k and circular composed of circular and scalloped lamination segments. The rotor lamination stack (T) hybrid rotor is levitated using only one set of stator poles. A Bs Flux density between stator pole k and scalloped second set of stator poles imparts torque to the scalloped portion k of the rotor, which is driven in a traditional switched reluctance rotor lamination stack (T) manner by a processor. Analysis was done for nonrotating rotor c Ek Energy density between stator pole k and circular poles that were oriented to achieve maximum and minimum 2 radial force loads on the rotor. The objective is to assess whether rotor laminations (J/m ) simple one-dimensional magnetic circuit analysis is sufficient for s Ek Energy density between stator pole k and preliminary evaluation of this machine, which may exhibit strong scalloped rotor laminations (J/m2) three-dimensional electromagnetic field behavior. Two magnetic c circuit geometries, approximating the complex topology of the Fk Magnetic force between circular lamination stack magnetic fields in and around the hybrid rotor, were employed in and stator pole k (N) formulating the electromagnetic radial force equations. s Reasonable agreement between the experimental results and the Fk Magnetic force between scalloped lamination theoretical predictions was obtained with typical magnetic stack and stator pole k (N) bearing derating factors applied to the predictions. Fm Magnetic force (N) Index Terms—Bearingless motor, electromagnetic device, F c Net downward magnetic force on circular rotor hybrid motor, magnetic field calculation, and switched- net reluctance motor.* lamination stack (N) s Fnet Net downward magnetic force on scalloped rotor I. NOMENCLATURE lamination stack (N) c c A0 Common cross-sectional area between a stator F Magnetic force on circular rotor lamination due 2 Tlower() pole and circular lamination stack (m ) to stator poles 1, 2, and 8 (N) s s 2 s Aa Average of Ab and Ac (m ) FTlower() Magnetic force on scalloped rotor lamination due s Ab Face area of stator pole 1 defined by length of to stator poles 1, 2, and 8 (N) scalloped segment, 2×10–4 m2 FT(net_lev) Total net magnetic levitating force on the rotor s s Ac Surface area inside scallop cavity, rθ L , (N) –4 2 4.35×10 m F c Total net magnetic levitation force on the Tnetlev()_ c circular rotor lamination due to stator poles 1, 2, Ak Common cross-sectional area between stator pole k and circular lamination stack (m2) 4, 5, 6, and 8 (N) s F s Total net magnetic levitation force on the Ak Common cross-sectional area between stator Tnetlev()_ pole k and scalloped lamination stack (m2) scalloped rotor lamination due to stator poles 1, 2, 4, 5, 6, and 8 (N) Manuscript received February 23, 2007. Ft(net) Total net downward magnetic radial force on Carlos R. Morrison is with the National Aeronautics and Space Administration, Glenn Research Center, Cleveland, OH 44135 U.S.A. (phone: rotor (N) 216–433–8447; fax: 216–977–7051; e-mail: [email protected]). s Mark W. Siebert is with the University of Toledo, Cleveland, OH 44135 Funet_ Net downward magnetic radial force on the U.S.A. (e-mail: [email protected]). unaligned scallop segment (Fig. 10) (N) Eric J. Ho was a summer intern at NASA Glenn Research Center and is now at University of Texas, Arlington, TX (email: [email protected]).*

1 s Ft(u_net) Total net downward magnetic radial force on R0 Nominal reluctance at the scalloped segment unaligned rotor (N) (1/H) c s FTupper() Magnetic force on circular rotor lamination due R5_ratio Reluctance ratio at pole 5 to stator poles 5, 4, and 6 (N) c R& Parallel reluctance at circular rotor lamination s FTupper() Magnetic force on scalloped rotor lamination due stack (1/H) to stator poles 5, 4, and 6 (N) s R& Parallel reluctance at scalloped lamination stack g General gap width (m) (1/H) ga Average stator-rotor gap spacing at stator pole 1 r Outer radius of rotor laminations (m) or 5 (Fig. 10) (m) v Flux volume between stator pole and rotor (m3) gb Gap length between a corner edge of the salient c Vk Flux volume between stator pole k and circular stator pole 1 (Fig. 10) and either of its adjacent rotor laminations (m3) rotor salient pole corner edges (m) s Vk Flux volume between stator pole k and scalloped gc Gap length between the midpoint of pole face 1 3 or 5 (Fig. 10) and the base of its adjacent scallop rotor laminations (m ) cavity (m) wm Magnetic energy (J) wc Magnetic energy between stator pole k and g Gap width at stator pole k (m) k k circular rotor laminations (J) –4 g0 Nominal gap width, 5×10 (m) s wk Magnetic energy between stator pole k and i Current (A) scalloped rotor laminations (J) x Rotor vertical displacement from nominal ik Current in coil k (A) c position (m) L Circular rotor lamination stack length, c 5.88×10–3 m φk Magnetic flux between stator pole k and circular laminations (Wb) Ls Scalloped rotor lamination stack length, φ s Magnetic flux between stator pole k and 1.94×10–2 m k scalloped laminations (Wb) –2 ltw Tooth width, 1.03×10 m φk Magnetic flux at stator pole k (Wb) –7 n Number of turns in coil, 215 μ0 Permeability of free space, 4π×10 H/m

Pk Permeance at stator pole k (H) II. INTRODUCTION Pc Permeance between stator pole k and the circular k There is a need for reliable, fail-safe, robust, compact, low- rotor lamination stack (H) cost electric motors for applications with high temperatures or s Pk Permeance between stator pole k and the extreme temperature variations; switched-reluctance motors scalloped rotor lamination stack (H) possess these characteristics [1]. These motors have been evaluated as high-speed starter-generators [2] and [3]. P Sum of permeances (H) T However, conventional switched-reluctance motors can suffer c PT Sum of permeances at the circular laminations from undesired vibration due to (1) unbalanced lateral forces (H) on the rotor caused by electrical faults, (2) mechanical offset s of the rotor, and (3) uncontrolled pulsed current in one or PT Sum of permeances at the scalloped laminations (H) more shorted coils [4]. A viable solution for mitigating mechanical vibration is to suspend the rotor magnetically via c Rk Reluctance between stator pole k and circular self-levitation. In addition, magnetic suspension of the rotor lamination stack (1/H) allows the motor to operate at much higher rotational s frequency for a prolonged period. This benefit is due largely to Rk Reluctance between stator pole k and scalloped lamination stack (1/H) the elimination of friction, as there is no physical contact between the stator and rotor. Consequently, motors c RT Total reluctance at circular rotor lamination stack incorporating magnetic bearings perform at higher efficiency (1/H) than motors incorporating ball bearings. Self-levitation also s obviates the need for a lubrication system, which has the RT Total reluctance at aligned rotor scalloped lamination stack (1/H) added benefit of significantly decreasing the weight and s complexity of turbomachinery mechanisms. RT1 Total reluctance at scallop cavity through pole 1 (Fig. 10) (1/H)

2 Methods for simultaneously levitating and rotating a rotor proportional-derivative (P-D) algorithm was used to levitate within a single stator in switched-reluctance motors have been the nonrotating rotor. The rotor was then vertically displaced proposed in [5] and [6]. In these motors, the technique of using a fixture containing a load cell, and the stator coil using differential stator windings was employed. The studies currents and force exerted on the rotor were recorded for each in [5] and [6] center primarily around the use of a main four- displacement value. In the corresponding analysis, the pole winding to rotate an eight-pole rotor while utilizing a measured currents were utilized directly, thus making the two-pole winding to apply radial force to the rotor with all of calculations independent of the control law. The question this the 12 stator poles having both windings thereon. A variation paper now seeks to answer is, can one successfully apply one- on this theme was described in [7], wherein only a single coil dimensional magnetic field analysis to the complex three- on each stator pole (in a 12/8 stator-rotor pole configuration) dimensional hybrid rotor to obtain the associated was employed to achieve motor-bearing action. electromagnetic radial force? Self-levitation (also called self-bearing) of motors has been A. Derivation of General Magnetic Force Equation achieved for nearly every type of electric motor, but is very marginal in performance for switched-reluctance motors with Fig. 2 depicts a portion of the rotor and a portion of a stator low numbers of poles. The motoring technique disclosed in [8] tooth on which the stator coil is energized to produce a will simultaneously levitate and rotate a rotor, not only for magnetic field flux density B. The magnetic field interacts (18/12) and (12/8) stator-rotor pole combinations, but also with the rotor at a distance g. The magnetic energy wm is thus (8/6) and (6/4) configurations while employing a single set of given by the volume integral equation coils positioned on each stator pole. The motoring techniques described in [5] to [7] are not applicable to motors having low 1 2 wBdvm = ∫∫∫ (1) stator-rotor pole configuration of (8/6) or (6/4) because of the 2μ0 dearth of stator-rotor poles in appropriate positions to apply levitating forces. The hybrid rotor technique described in [8] where μ0 is the permeability of free space. Integration over the will assure robust bearingless operation in all four gap volume is possible if the magnetic core material is aforementioned stator-rotor pole configurations. With circular operating within its linear range so that the energy stored in laminations on the shaft, levitating force is always assured as the core can be neglected compared with that in the gap. the rotor spins. c Substituting dv= L ltw dg in (1), and neglecting leakage A disassembled eight-pole stator, six-pole hybrid rotor and fringing, yields (motor) is shown in Fig. 1. The rotor lamination stack has a cylindrical portion of its length (indicated by the right arrow in B2 the picture) used for levitation and a scalloped portion c wLldgmtw= ∫ (2) (indicated by the left arrow) used for both levitation and 2μ0 motoring. The currents in the coils of the eight-pole stator are c controlled by an algorithm that handles both the levitating and where L is the stack thickness of the circular laminations, ltw motoring functions. During normal operation, two pairs of is the width of a stator tooth, and dg is the differential gap opposing stator poles (at right angles to each other) levitate the spacing. Taking the derivative of wm with respect to g rotor. The remaining two pairs of stator poles exert torque on produces the magnetic force relation the six-pole rotor lamination stack to produce rotation. The relative lengths of the circular and multipole lamination stacks dw B2 F =−m =− ⋅Llc . (3) on the rotor can be chosen to tailor the performance of the mtw dg 2μ0 motor for a specific application. For a given overall length, increasing the length of the multipole stack relative to the The mathematical form and value of the B field impinging circular stack results in an increase in torque relative to on the asymmetric rotor will be determined next. levitation load capacity and stiffness, and vice versa. For this –2 paper, the rotor has a 2.525×10 -m- (1-in.-) long hybrid III. NONLEVITATED MAGNETIC EQUIVALENT CIRCUIT –3 segment comprising a 5.883×10 -m- (0.233-in.-) long section MODELING FOR CENTERED ROTOR –2 of circular laminations for levitation and a 1.937×10 -m- Fig. 3 is an axial view of the motor. (0.767-in.-) long portion having six salient poles for rotation. This paper focuses exclusively on the nonrotating magnetic A. Circular Lamination Force Contribution bearing action forces in the (8/6) hybrid motor. Two types of A complex three-dimensional magnetic field topology magnetic force measurements were made for the nonrotating exists in and around the rotor because of its hybrid design. rotor in which two diametrically opposed rotor poles were However, the analysis will be restricted to a one-dimensional aligned with stator poles. In the first measurement, only one approach in order to reduce the mathematical complexity of stator pole was energized, and the force on the centered rotor the static radial force derivation. Each rotor segment (circular was recorded as a function of current. In this situation, the flux lamination stack and scalloped lamination stack) will be through the excited pole returns through a number of treated as an independent entity, and accordingly, we develop unexcited poles. In the second type of measurement, a a one-dimensional magnetic radial force description for each

3 rotor segment. The magnetic radial forces will then be c 8 ⎛⎞g summed to obtain the total static magnetic radial force on the R = ⎜⎟0 . (11) T 7 ⎜⎟c rotor. ⎝⎠μ00A Fig. 4 depicts the equivalent two-dimensional flux distribution in the rotor’s circular laminations and the eight From the basic magnetic circuit equation, the magnetomotive force stator poles. Each stator pole is at a distance g0 (gap width) from the circular lamination stack and each pole shares a mmf= φ R , (12) c common cross-sectional area A0 with the circular laminations. Fig. 5 is the corresponding magnetic equivalent and noting that mmf = ni, where n is the number of turns in the circuit of Fig. 4. Both figures will be employed in deriving the coil and i is the current in the coil, we can now write the flux static magnetic radial force equations. expression

The common cross-sectional area Ac between the circular 0 ni lamination stack and each stator pole can be expressed as φ c = . (13) 1 c RT cc AAkk ==0 , 1...8 (4) Substituting (11) into (13) yields

cc c where A0 = Lltw . The flux at stator pole 1 φ1 , produced by 7 ⎛⎞μ Acni energizing coil 1, is distributed equally among the other seven φ c = ⎜⎟00 . (14) 1 ⎜⎟ poles. The flux distribution obeys the relation 8 ⎝⎠g0

8 The flux density at pole 1 can thus be written as φ cc= φ , (5) 1 ∑ k k=2 φ c 7 ⎛⎞μ ni Bc ==1 0 . (15) 1 c ⎜⎟ and for a rotor that is radially centered, the reluctances A0 8 ⎝⎠g0 between the stator poles k and the rotor are related by Utilizing (3), we can express the magnetic radial force as cc RRkk==+1, k 1...7 . (6) 1 ⎛⎞Ac 2 FBcc= ⎜⎟0 (16) The total reluctance for the flux passing through the powered 11⎜⎟() 2 μ0 pole is ⎝⎠

ccc for a uniform B field, hence RRRT =+1 & (7) c 222 149⎛⎞⎛⎞⎛A μ ni ⎞ c F c = ⎜⎟⎜00 ⎟. (17) where R1 is in series with the parallel reluctances 1 ⎜⎟⎜⎟⎜2 ⎟ c c c c c c c c 264⎝⎠⎝⎠⎝μ0 g0 ⎠ R2 , R3 , R4 , R5 , R6 , R7 , and R8 . The parallel reluctance R& is Equation (17) is the static magnetic radial force due to pole 1 c (Fig. 3). The oppositely directed radial forces produced by c R1 R& = , (8) poles 3 and 7 will sum to zero. Likewise, the resultant 7 magnetic force produced by poles 2, 4, 6, and 8 is zero. Hence, only the radial force at pole 5 will contribute to the resultant and substituting (8) into (7), produces force on the rotor. Because of the flux symmetry in the rotor and stator, we can write the magnetic field at pole 5 cc8 RRT = 1 . (9) 7 cc1 BB51= , (18) 7 For a rotor-stator gap g0 , and substituting (15) into (18) yields g Rc = 0 (10) 1 c c 1 ⎛⎞μ0ni μ00A B = ⎜⎟. (19) 5 8 g ⎝⎠0 hence, (9) can be rewritten as The magnetic radial force at pole 5 obeys the relation

4 1 ⎛⎞Ac 2 and FBcc= ⎜⎟0 (20) 552 ⎜⎟μ () ⎝⎠0 sss11 AAAk210k ==, = 1...4 . (26) 33 hence The reluctance between the rotor and stator poles for the radially centered rotor is hence given as ⎛⎞⎛Ac 222ni ⎞ c 11⎛⎞00μ F5 = ⎜⎟⎜⎟⎜ ⎟. (21) 264⎝⎠⎜⎟⎜μ g 2 ⎟ ⎝⎠⎝0 0 ⎠ s ggg000ss3 RR12====, 3 R 1 (27) μμAAAsss 01μ 1 01 The net downward magnetic force produced by poles 1 and 5 0 3 on the circular laminations is given as

and ccc Fnet =−FF15. (22) ss s s RR15=== and R 2k R 2 , k 2...4 . (28) Substituting (17) and (21) into (22) yields the net downward magnetic force on the circular lamination stack, The total reluctance in the magnetic circuit can be written as

⎛⎞c 2 c 3 An00μ 2 s ss Fnet = ⎜⎟i . (23) RRRT = 1 + & (29) 8 ⎜⎟g 2 ⎝⎠0 s where R1 is in series with the parallel reluctances Substituting the actual value of the constants gives s s s s s R2 , R4 , R5 , R6 , and R8 . The resultant parallel reluctance is thus obtained by c 2 Fnet = 5.037 i . (24) 114 1 We now determine the static magnetic radial force on the =+ . (30) RRs ss∑ R scalloped laminations by employing some aspects of the & 52k=1 k previous discussion. Substituting (27) and (28) into (30) yields B. Scalloped Lamination Force Contribution Fig. 6 depicts the equivalent two-dimensional flux 3Rs Rs = 1 (31) distribution in the rotor scalloped laminations and their & 7 adjacent stator poles. Each rotor pole is at a nominal distance

g0 from its adjacent stator poles. Four stator poles—2, 4, 6, and substituting (31) into (29) produces and 8 (Figs. 3 and 6)—share a common cross-sectional area 1 As with four adjacent rotor poles. Fig. 7 is the 3 0 s 10 s corresponding magnetic equivalent circuit of Fig. 6. RRT = 1 , (32) 7 Additionally, in the spirit of rough approximation, we neglect any fringing effects from the rotor to stator poles 3 and 7. The and since expected large reluctance between the aforesaid poles and their scalloped cavities will result in a relatively small return s g0 flux. However, in later sections analyzing the unaligned R1 = , (33) μ As centered rotor and the aligned levitated rotor, the fringing 01 effect will be included to give a more complete description of the levitation force on the rotor. (32) can be rewritten as Observation of Fig. 3 reveals that only the stator poles labeled as 1, 2, 4, 5, 6, and 8 provide significant paths for the 10 ⎛⎞g Rs = ⎜⎟0 . (34) circulating magnetic flux, and each pole 2, 4, 6, and 8 has an T ⎜⎟s 7 ⎝⎠μ01A equivalent, effective common stator-rotor area that is one-third that of stator poles 1 or 5. The common cross-sectional areas Using the magnetomotive force argument, we can derive (35) between the rotor scalloped lamination stack and a stator poles can be expressed as ss φ1 RniT = (35) As ==AAss (25) 150 from which we obtain the relation

5 ni s s s 3 ⎛⎞μ A ni φ1 = . (36) φ = ⎜⎟05 . (45) Rs 5 ⎜⎟ T 10 ⎝⎠g0

Substituting (34) into (36) yields The flux density at pole 5 is thus given as

7 ⎛⎞μ Asni s φ s = ⎜⎟01 . (37) s φμ503 ⎛⎞ni 1 ⎜⎟ B5 ==⎜⎟, (46) 10 g0 s ⎝⎠ A5 10 ⎝⎠g0

The flux density extant at pole 1 and its adjacent salient rotor and the magnetic force between pole 5 and its adjacent salient pole is thus given by rotor pole is given by

φ s 7 ⎛⎞μ ni s s 1 0 s 1 ⎛⎞A s 2 B1 ==⎜⎟. (38) FB= ⎜⎟5 . (47) As 10 g 55⎜⎟() 1 ⎝⎠0 2 ⎝⎠μ0

The magnetic force is represented as Substituting (46) into (47) yields

⎛⎞s 2 s 1 A1 s 19⎛⎞⎛As μ 222ni ⎞ FB11= ⎜⎟ , (39) s ⎛⎞50 ⎜⎟() F5 = ⎜⎟⎜⎟⎜ ⎟. (48) 2 ⎝⎠μ0 ⎜⎟⎜2 ⎟ 2⎝⎠ 100 ⎝⎠⎝μ0 g0 ⎠ and substituting (38) into (39) engenders the expression The net downward magnetic force produced by poles 1 and 5 on their corresponding salient rotor poles is thus given by 149⎛⎞⎛⎞As ⎛⎞μ 222ni F s = ⎜⎟1 ⎜⎟0 . (40) 1 ⎜⎟⎜⎟⎜⎟2 s ss 2⎝⎠ 100 ⎝⎠μ0 ⎝⎠g0 Fnet =−FF15. (49)

Equation (40) represents the magnetic force, due to stator pole Substituting (40) and (48) into (49) yield the force expression 1, on the aligned salient pole of the rotor. The resultant magnetic force on the rotor, produced by poles 2 and 8, ⎛⎞s 2 s 1 An00μ 2 cancels the resultant magnetic force produced by poles 4 and Fnet = ⎜⎟i (50) 5 ⎜⎟2 6. Hence, only the magnetic force at pole 5 will contribute to ⎝⎠g0 the resultant force on the rotor. The flux at pole 5 can be expressed as where Ass= Ll . Equation (50) represents the net downward 0 tw magnetic force on the scalloped lamination stack. Substituting 4 φ s =−φφss. (41) the actual values of the constants gives 51∑ 2k k=1 s 2 Fnet = 8.845 i . (51) The flux relation can be written as The total net downward magnetic force on the centered φφss==,k 2...4 , (42) nonlevitated hybrid rotor is the sum of the net downward 22k magnetic force on the circular lamination stack and the net

downward magnetic force on the scalloped lamination stack: and substituting (42) into (41) results in the simplified expression cs Ft() net=+FF net net . (52) s ss φ51=−φφ4 2. (43) Substituting (24) and (51) into (52) yields Flux symmetry consideration allows us to write the relation 2 Ftnet()= 13.88 i . (53) 1 φ s = φ s , (44) 217 The fraction of force contributed by the circular lamination segment is and substituting (44) into (43) yields

6 F c 5.037 i2 nonlevitated unaligned rotor-stator radial force equations are net == 0.36 36% , (54) 2 derived for the hybrid rotor. Ftnet() 13.88 i E. Nonlevitated Unaligned Rotor Analysis and the fraction by the scalloped segment is The analysis presented in previous sections clearly demonstrates that one can make a reasonably accurate s 2 prediction of electromagnet forces on a radially centered Fnet 8.845 i == 0.64 64% . (55) hybrid rotor using two-dimensional visualization and one- F 2 tnet() 13.88 i dimensional electromagnetic force analysis. With this fact established, we will estimate the radial force on the unaligned Note that the circular lamination stack experiences radially centered rotor, that is, a rotor that is rotated so that the approximately 36% of the total net downward magnetic force, motoring section has an unfavorable misalignment with the while the scalloped lamination experience approximately 64% powered stator pole 1, as depicted in Fig. 10. Using a force of said force. The stack lengths of the circular and scalloped c s symmetry argument similar to that presented in earlier laminations are L = 0.23 and L = 0.77, respectively. Note that sections, it can be shown that only poles 1 and 5 are relevant the forces in (54) and (55) are not proportional to the stack in determining the net force on the hybrid rotor. The analysis lengths. This discrepancy is due to the differences in return begins by approximating the reluctance between poles 1 or 5 paths of the flux in the hybrid rotor. The following section and the adjacent scallop cavities: outlines the experimental procedure for obtaining the nonlevitated magnetic radial force on the rotor. s ga s C. Experimental Centered-Force Procedure RR15== (57) μ0 Aa The experimental nonlevitated magnetic radial force on the rotor was obtained by mounting the rotor between two load g + g cells (see Fig. 8) while ensuring that the rotor’s axis was ⎛⎞bc where ga = ⎜⎟ is the average stator-rotor gap spacing radially centered in the stator. A salient rotor pole was aligned ⎝⎠2 with stator pole 1 (see Fig. 3). The preload voltages of the at stator poles 1 or 5 (Fig. 10), gb is the distance between a cells were then zeroed to eliminate the preload weight from corner edge of the salient stator pole 1 or 5 and either of its the data. Stator pole 1 was energized, and the resulting adjacent salient rotor pole corner edges, and gc is the gap magnetic force on the rotor produced a larger reading on the length between the midpoint of either pole face and the base of lower cell and a smaller reading on the upper cell. The ⎛⎞As + As its adjacent scallop cavity. Here, A = ⎜⎟bc is the difference in the readings gave the net force on the rotor. This a ⎜⎟2 procedure was repeated for several values of coil current ⎝⎠ s ranging from 0.982 to 2.027 A. The procedure was repeated average of the stator pole face area Ab and its adjacent scallop with pole 5 as the sole energized pole. s cavity area Ac . The face area of pole 1 is D. Experimental and Theoretical Centered-Force AAss==×210−42 m and the surface area inside a scallop Comparison b 0 s −42 The experimental and theoretical magnetic force data are cavity was determined to be Ac =×4.35 10 m . presented in Table I, and the related magnetic force plots are Substituting these values into the average stator-rotor gap –3 shown in Fig. 9. Because of flux leakage and field fringing spacing and area expressions result in ga = 6.25×10 m and effects at the stator and rotor pole edges, uncertainties in –4 2 s Aa = 3.17×10 m . We can rewrite Aa in terms of A0 : geometry, rotor-stator alignment, and material properties, we must apply a derating factor to (53). Reference [9] presents a A detailed treatment of the derating factor, wherein a factor of a = 1.59,A =× 1.59 As , (58) s a 0 0.88 was determined for one particular active magnetic A0 bearing. In contrast, a graph in [10] shows a derating factor of –4 0.6 for our gap g0 (5×10 m). This rotor’s configuration is, of and ga in terms of g0 : course, different from that of a pure magnetic bearing. A derating of 0.87 gives the closest agreement with the g a = 12.5,g =× 12.5 g . (59) experimental results obtained for the hybrid rotor (see (56) and g a 0 Fig. 9): 0

2 Substituting these values into (57) yields Ftnet()=×()13.88 0.87 i . (56) gg RRs ==a 7.860 = 7.86 s (60) Fig. 9 depicts the experimental least-squares-fit magnetic 10s μ0 Aa μ00A force curves for the upper and lower poles, and the theoretical magnetic force curve. In the following section, the

7 where Rs is the nominal reluctance at the scalloped segment. g 0 Substituting Rs ==×0 1.99 1061 H− into (69) gives Substituting the actual value of the constant yields 0 s μ00A

Rs =×1.63 1071 H− . The total flux at pole 1 has the form RRss==7.86 R s = 1.57 × 1071 H− . (61) T1 15 0

ni Here, the estimated reluctance at the unaligned pole is almost φ s ==6.15 × 10−8 H ni (70) 1 s () eight times that of an aligned pole. Note that this value of the RT1 reluctance applies to both poles 1 and 5. The remaining pole reluctances are from which we can determine the flux density:

gggg RRRRssss====0000,,,, φ s ⎛⎞6.15× 10−8 H 2468s sss Bniis ==1 ⎜⎟ =6.63 × 10−22 H/m . (71) μμμμ02A 04AAA 06 08 1 s ⎜⎟−42 () (62) A0 ⎝⎠210× m gg RRss==00, and . 37ss μμ03AA 07 The radial force at pole 1 is thus

We know that 1 ⎛⎞As 2 F ss==⎜⎟0 Bi0.35 2 . (72) 11⎜⎟() s ss 2 ⎝⎠μ0 A370==AA (63)

The next task is to determine the flux fraction at pole 5, hence which will allow computation of the magnetic force at pole 5.

The difference between this force and the force at pole 1 will s ss RRR370== (64) give the net magnetic force on the unaligned hybrid rotor. The fraction of the total flux passing through pole 5 is then determined by its reluctance ratio, the combined parallel reluctance of all the return poles divided by the reluctance of pole 5: s gg003 s RR20===3 (65) ⎛⎞As μ As 0 00 s s μ0 ⎜⎟ R 0.29R ⎜⎟3 s & 0 ⎝⎠ R5_ratio == 0.037 . (73) RRss7.86 50 where RRkss==, 2...4 . 22k The flux at pole 5 is now obtained by employing (70):

The total parallel reluctance due to poles 2, 3, 4, 5, 6, 7, and 8 ni is given as φφss==×R s0.037 (74) 55_1ratio s RT1 1421 =++, (66) s sss or simply R& RRR235

s −7 and substituting (61), (64), and (65) into (66) gives φ5 =×4.86 10 i . (75)

142 1 The force at pole 5 is =++ . (67) RRRs 37.86ss R s & 00 0 2 φ s s ()5 −42 F5 ==×4.71 10 i . (76) Equation (67) simplifies to 2μ As 00

s s RR& = 0.29 0 , (68) The net downward magnetic radial force on the unaligned scalloped section of the rotor due to poles 1 and 5 is hence, the total reluctance of the magnetic circuit through pole 1 is s ss Funet_ =−FF15. (77)

s ss s s s Substituting (72) and (76) into (77) produces RRRT11=+=& 7.86 R 0 + 0.29 R 0 = 8.15 R 0. (69)

8 s −42 A. Levitation Flux Derivation Fu_ net =−×()0.35 4.71 10 i (78) We designate the coil currents as i1, i3, i5, and i7 for the respective stator poles 1, 3, 5, and 7. Each coil has n turns. or more simply These currents generate the magnetic fluxes φ , φ , φ , φ , 1 3 5 7 s 2 φ2,8 , and φ4,6 , where φ2,8 and φ4,6 represent the return Fu_ net 0.35i . (79) fluxes due to pole pairs (2 and 8) and (4 and 6) on either side of pole 1 and pole 5, respectively. Referencing Fig. 11, (84) to The ratio of the net downward magnetic force on the (88) are written by applying the magnetic circuit equation to unaligned scalloped stack to that on the aligned scalloped various closed paths (all of which go through pole 1, which stack F s is net has a permeance P1 ):

s F 0.35i2 φ φ u_ net == 0.039 3.9% . (80) 3 − ni=+1 ni (84) s 2 31 Finet 8.85 PP31

Note that the unaligned scallop magnetic force load is only φ5 φ1 3.9% of the magnetic force load produced by the aligned + ni51=+ ni (85) PP scallop. This is a reasonable result, considering the large 51 reluctances of the scallop cavities located at poles 1 and 5. Taking into account the contributions of the circular φ φ 7 − ni=+1 ni (86) laminations, the total net downward magnetic radial force on 71 PP71 the unaligned rotor is

cs φ2,8 φ1 FFFt(_ u net )=+ net u _ net . (81) + 0 =+ni1 (87) PP2,8 1

We know from (24) and (51) that the aligned net radial φ φ force on the circular laminations and scallops are 4,6 + 0 =+1 ni . (88) c 2 s 2 PP1 Fnet 5.037i and Fnet 8.845 i , respectively, from which 4,6 1 the total net downward magnetic force on the aligned rotor is Conservation of the magnetic flux assures that the sum of the cs magnetic fluxes is equal to zero, hence Ft() net=+FF net net . (82)

φφφφ+ +++ φ + φ =0 . (89) The ratio of the total net downwards magnetic force of the 13572,84,6 unaligned rotor to the total net downwards magnetic force on the aligned rotor is The solution of (84) to (89) is given by

2 P Ftu(_ net ) (5.037+ 0.348)i φ =−1 ⎡Pi − Pi + Pi == 0.39 39% . (83) 1335577⎣ 2 PT (90) Ftnet() (5.037+ 8.845)i ++++()PPPP3572,84,61 + Pin⎦⎤ Equations (80) and (83) highlight the importance of the circular lamination stack; that is, the load capacity in the P φ =+−3 ⎡Pi Pi P i unaligned orientation is substantially enhanced by the circular 3115577P ⎣ lamination segment. In subsequent sections, the general T (91) levitation force equations are derived for the hybrid rotor that ++++()PPP1572,84,63 P + Pin⎦⎤ is displaced vertically from its radially centered position within the stator. P φ =−5 ⎡Pi − Pi + Pi 5331177P ⎣ IV. LEVITATION MAGNETIC EQUIVALENT CIRCUIT MODELING T (92) The analysis begins by employing the magnetic equivalent ++++()PPP1372,84,65 P + Pin⎦⎤ circuit depicted in Fig. 11, which relates to the magnetically levitated state of the nonrotating, aligned, vertically displaced P φ =−+7 ⎡Pi Pi Pi hybrid rotor. The rotor was magnetically levitated using stator 7113355⎣ PT (93) poles 1, 3, 5, and 7 (Fig. 3). ++++()PPPP1352,84,67 + Pin⎦⎤

9 P c ccccc c 2,8 PT ≡++++ PPPPP13572,84,6 + P. (103) φ2,8=−+−[]Pi 1 1 Pi 3 3 Pi 5 5 P 7 i 7 n (94) PT The flux at pole 5 is obtained by utilizing (92): P φ =−+−4,6 Pi Pi Pi P i n (95) 4,6[] 1 1 3 3 5 5 7 7 Pc PT φ cccc=−5 ⎡Pi − Pi + Pi 5331177c ⎣ PT (104) where for notational convenience we define the sum of +++++PPPPcccc Pin c ⎤ . permeances as ()1372,84,65⎦

PPPPPPT ≡++++13572,84,6 + P. (96) The equations for the magnetic field, energy density, flux volume, and energy at pole 5 are, Equations (90), (92), (94), (95), and (96) are used to derive the electromagnetic levitation force equations for the rotor c c φ5 ()x displaced vertically inside the stator. In subsequent sections, Bx5 ()= (105a) Ac separate magnetic force equations for the circular and 0 scalloped lamination segments are derived. [()]Bxc 2 B. Levitation Force on the Circular Laminations Exc ()= 5 (105b) 5 2 The common cross-sectional area between the circular μ0 cc laminations and pole 1 or 5 is given as A0 = Lltw . The rotor’s cc vertical displacement x from its nominal position affects the Vx505()= Agx () (105c) gap widths gk, hence

wxccc()= V () xEx () (105d) g =+gx (97) 555 10

respectively. The magnetic force on the rotor due to pole 5 is at stator pole 1, and then obtained by differentiating the magnetic energy wc with 5 respect to the rotor displacement, hence g =−gx (98) 50 cc ccdw55() x dV () x at stator pole 5. In addition, the gaps at the return flux poles 2 FEx55=− =− () . (106) and 8 can be written as dx dx

ggx=+cos(45 ° ) (99) Using (95), the return magnetic flux equation for poles 4 and 6 2,8 0 is and for poles 4 and 6 Pc φ ccccc=−+−4,6 Pi Pi Pi Pi n (107) 4,6c () 1 1 3 3 5 5 7 7 ggx4,6=− 0 cos(45 ° ) (100) PT

c where 45° is the angle between adjacent poles. The where PT is the sum of permeances at the circular approximate permeances between the poles and the circular laminations. The equations for the magnetic field, energy lamination stack are density, flux volume, and energy at poles 4 and 6 are,

μ Ac c ccc00 c φ4,6 ()x PPP370=== (101) Bx()= (108a) g 4,6 c 0 A0

μμAAcc2 μ A c 2 PPPccc==00,, 00 = 00 , ⎡⎤Bxc () 152,8 c ⎣⎦4,6 gg15 g 2,8 Ex()= (108b) (102) 4,6 c 2μ0 c 2μ00A and P4,6 = , g4,6 cc Vx4,6()= 2 Agx 0 4,6 () (108c) and applying (96) to the circular laminations, ccc wxVxEx4,6()= 4,6 () 4,6 () (108d)

10 respectively. The magnetic force due to poles 4 and 6 is Here, 51.6° is the angle between the midpoint of stator pole 1 and either of the midpoint of the overlap of its adjacent poles 2 dwcc() x dV () x or 8 with their adjacent rotor poles, and similarly for stator FExcc=−4,6 =− () 4,6 . (109) pole 5 and its adjacent poles 4 or 6. The approximate fringing 4,6dx 4,6 dx permeance Ps or Ps at pole 3 or 7, respectively, is similar in 3 7 The total magnetic force due to pole 5 and its adjacent poles 4 value to that used in (70), hence and 6 is thus ss1 −8 PP37== =6.1509 × 10 H . (117) ccc Rs FTupper()54,6=+FF. (110) T1

Similarly, we can calculate the net downward magnetic force The remaining permeances are created by poles 1, 2, and 8. The magnetic force at pole 1 is ss s μμAA2μ00(2,8)A PPPsss==00,, 00 = , dwcc() x dV () x 152,8 FExcc=−11 =− () , (111) gg15 g 2,8 11dx dx (118) 2μ As s 00(4,6) and P4,6 = . and the magnetic force due to the return flux poles 2 and 8 is g4,6 thus given as The sum of the permeances as defined by (96) is cc ccdw2,8() x dV 2,8 () x F2,8=− =−Ex 2,8 () s. (112) s sssss s dx dx PT ≡++++ PPPPP13572,84,6 + P. (119)

The total magnetic force due to pole 1 and its adjacent poles 2 Using (92), the equation for the flux at pole 5 is and 8 is thus

Ps ccc ssss5 ⎡ FTlower()12,8=+FF. (113) φ5331177=−Pi − Pi + Pi Ps ⎣ T (120) The net magnetic levitation force on the circular laminations, ssss s ⎤ +++++()PPPP1372,84,65 Pin. due to poles 1, 2, 4, 5, 6, and 8 can now be written: ⎦

The equations for the magnetic field, energy density, flux FFFccc=+. (114) T(_) net lev T ( upper ) T ( lower ) volume and energy at pole 5 are, respectively,

Note that for the vertically displaced rotor, the net force on the φ s circular lamination stack, due to poles 3 and 7, is identically Bs = 5 (121a) 5 s zero. A0 C. Levitation Force on the Scalloped Laminations s 2 We now determine the net force on the scalloped s [()]Bx5 Ex5 ()= (121b) laminations. The common cross-sectional area between the 2μ0 ss scalloped laminations and poles 1 or 5 is given as A0 = Lltw , where Ls is the scalloped rotor lamination stack length. For the ss Vx505()= Agx () (121c) return poles 2 or 8 and 4 or 6, the common cross-sectional areas are ALlss= and ALlss= respectively, (2,8)tw (2,8) (4,6)tw (4,6) sss where the corresponding tooth widths are wx555()= V () xEx () (121d)

1.03× 10−2 The magnetic force on the scalloped laminations, due to pole lxtw(2,8) =−°sin(51.6 ) (115) 5, is thus obtained by differentiating the magnetic energy 3 s wx5 () with respect to the rotor’s displacement, thus and ss ssdw55() x dV () x −2 FEx=− =− () . (122) 1.03× 10 55dx dx lxtw(4,6) =+°sin(51.6 ) . (116) 3 Using (95), the return magnetic flux equation for poles 4 and 6

11 is The net magnetic levitation force on the scalloped laminations due to poles 1, 2, 4, 5, 6, and 8 is thus Ps φ sssss=−+−4,6 Pi Pi Pi Pi n. (123) sss 4,6s () 1 1 3 3 5 5 7 7 FFFT(_) net lev=+ T ( upper ) T ( lower ). (130) PT

Note that for the vertically displaced rotor, the net force on the The equations for the magnetic field, energy density, flux scalloped lamination stack due to poles 3 and 7 is identically volume, and energy at poles 4 and 6 are, respectively, zero. The total net magnetic levitating force on the vertically

displaced hybrid rotor can now be written: s φ Bs = 4,6 (124a) 4,6 s cs A4,6 FFFTnetlev(_)=+ Tnetlev (_) Tnetlev (_). (131)

s 2 The experimental measurements and their comparisons with s [()]Bx4,6 Ex4,6 ()= (124b) the predictions of (131) are presented in the following 2μ0 sections.

D. Experimental Levitation Force Procedure ss Vx4,6()= 2 Agx 4,6 4,6 () (124c) A schematic of the experimental setup is displayed in Fig. 12. The hybrid rotor was configured as shown in Fig. 3 and sss then magnetically levitated, without rotation, using stator wxVxEx4,6()= 4,6 () 4,6 () (124d) poles 1, 3, 5, and 7 and a standard P-D control law. A vernier

device, supporting a load cell, effects vertical rotor motion. and the equation for the magnetic force due to poles 4 and 6 is The rotor was then preloaded to ensure tension in the load cell now given as for both positive and negative vertical displacements and the load cell voltage zeroed, thereby eliminating the combined dwss() x dV () x ss4,6 4,6 preload and rotor weight from the data. Fifteen position data FEx4,6=− =− 4,6 () (125) dx dx points, along with their four coil currents i1, i2, i3, and i4, were recorded. where ggx=−cos(51.6 ° ) and 51.6° is the angle between 4,6 0 E. Experimental and Theoretical Levitation Force the midpoint of stator pole 5 and either of the midpoint of the Comparison overlap of its adjacent poles 4 or 6 with their adjacent rotor The experimental rotor displacements along with their pole. The equation for the total magnetic force due to pole 5 associated coil currents and the levitation force equations and its adjacent poles 4 and 6 on the scalloped laminations is derived above were programmed into Mathematica (Wolfram thus Research), which generated the predicted force data points.

s ss The absolute values of the predicted force data (obtained from FTupper()54,6=+FF. (126) (131)) and the experimental force data were then plotted using Excel (Microsoft Corporation). Because of (1) flux leakage Similarly, we can calculate the net magnetic force created by and field fringing effects at the stator and rotor pole edges, poles 1, 2, and 8. The magnetic force at pole 1 is (2) experimental errors related to the forced rotor displacement, and (3) uncertainties in rotor-stator geometry, ss alignment and material properties, a derating must be applied ssdw11() x dV () x FEx11=− =− () , (127) to (131). For this data set, a derating of 0.72 gave the closest dx dx agreement to the experimental result (see Table II and

Fig. 13). and the magnetic force due to the return flux poles 2 and 8 is thus given as V. CONCLUDING REMARKS ss The electromagnetic radial forces within a novel eight- ssdw2,8() x dV 2,8 () x FEx2,8=− =− 2,8 () . (128) stator-pole, nonrotating six-rotor-pole bearingless switched- dx dx reluctance motor were examined theoretically and experimentally. Each axial rotor segment (circular lamination The total magnetic force due to pole 1 and its adjacent poles 2 stack and scalloped lamination stack) was treated as an and 8 is hence independent entity, and one-dimensional magnetic force

equations were developed for each. The total net magnetic s ss FTlower()12,8=+FF. (129) forces on the rotor were obtained by summing the net magnetic forces on each segment and then applying derating

12 factors of 0.87 for the nonlevitated case and 0.72 for the Carlos R. Morrison was born in levitated case. In addition, it was demonstrated analytically Kingston, Jamaica, West Indies. He that the load capacity of the rotor in the unaligned orientation received his B.S. (Hons.) degree in Physics is substantially enhanced by the presence of a circular with a Mathematics minor in 1986 from lamination segment (this is critical for maintaining rotor Hofstra University, Hempstead Long Island, stability during the motoring operation). Based on the NY, and an M.S. in Physics in 1989 from analysis, we conclude that two-dimensional visualization and Polytechnic University, Brooklyn, NY. one-dimensional magnetic field analysis can be used for In 1989, he joined the NASA Glenn estimating electromagnetic levitation forces on the hybrid Research Center, Cleveland, OH, as a staff scientist in the rotor with a derating factor of about 0.8 applied to the Solid State Physics branch, and in 1999 he transferred to the calculated force. Structural Dynamics and Aeroelasticity branch. He has authored and coauthored several journal and technical articles ACKNOWLEDGMENT associated with aerospace and electromagnetic devices. He The novel rotor (motor) was developed with support from holds patents on the “Morrison Motor” and software the NASA “Strategic Research Fund” (SRF) Combined technologies used to control magnetic bearings. He is Motor/Magnetic Bearing Project (Dr. Marvin Goldstein, currently engaged in research on switched-reluctance motors manager), the “Revolutionary Aeropropulsion Concept” and magnetic shape memory alloys. Mr. Morrison is a member (RAC) Project, High Power Density Motors for Aircraft of the American Physical Society and the National Technical Propulsion (Mr. Leo Burkardt, manager) and subsequent Association. support from the “Alternate Fuel Foundation Technology” (AFFT), with Mr. Dave Ercegovic as project manager. Mark Siebert was born in Toledo, OH, in 1967. He received Special recognition is extended to Dr. Gerald V. Brown for his B.S. in Mechanical Engineering from the University of Cincinnati in 1991 (magna cum laude) and an M.S. in his many insightful discussions that ultimately led to the Mechanical Engineering from the University of Toledo in theoretical analysis presented in this paper, Mr. Ben Ebihara 1993. and Mr. Carl Buccieri for their contributions in fabricating the Mr. Siebert currently works for the University of Toledo at motor, and Mr. Joseph Wisniewski and Mr. Gerald Buchar for the NASA Glenn Research Center. Mr. Siebert conducted their assistance with the motor’s electrical and electronics research in the areas of passive magnetic bearings, cryogenic wiring. switched-reluctance motors, self-bearing switched-reluctance motors, solid lubricants, and space experiments. Mr. Siebert REFERENCES also conducted the ground testing for the Mars Pathfinder [1] G. Montague, et al., “High-temperature switched-reluctance electric wheel abrasion experiment on the Sojourner rover and worked motor,” NASA Tech Briefs, Feb. 2003. [2] E. Richter and C. Ferreira, “Performance evaluation of a 250 kW on a followup experiment to measure the magnitude of switched reluctance starter generator,” IEEE IAS’95, 1995, pp. electrostatic charging of the rover. 434−440. [3] S.R. MacMinn and W.D. Jones, “A very high speed switched-reluctance starter-generator for aircraft engine applications,” Aerospace and Eric J. Ho was born in Fort Worth, TX. Electronics Conference, IEEE’ 89, 1989, pp. 1758−1764. He is currently studying at the University of [4] T.J.E. Miller, “Faults and unbalance forces in the switched reluctance Texas at Arlington towards his B.S. degree machine,” IEEE Trans. Ind. Applicat., vol. 31, pp. 319−328, Mar./Apr. in Electrical Engineering. 1995. [5] M. Takemoto, K. Shimada, A. Chiba, and T. Fukao, “A design and In 2006, he joined the NASA Glenn characteristics of switched reluctance type bearingless motors,” in Proc. Research Center in Cleveland, OH, as a 4th Int. Symp. Magnetic Suspension Technology, NASA/CP⎯1998- summer intern in the Structural Dynamics 207654, May 1998, pp. 49−63. and Aeroelasticity branch, and worked [6] M. Takemoto, H. Suzuki, A. Chiba, T. Fukao, and M.A. Rahman, “Improved analysis of a bearingless switched reluctance motor,” IEEE closely under the supervision of Mr. Carlos Morrison. Mr. Ho Trans. Ind. Applicat., vol. 37, Jan./Feb. 2001. is a student member of the Institute of Electrical and [7] M.A. Preston, J.P.F. Lyons, E. Richter, and K. Chung, “Integrated Electronics Engineers. magnetic bearing/switched reluctance machine,” U.S. Patent 5,424,595,

June 13, 1995. [8] C.R. Morrison, “Bearingless switched reluctance motor,” U.S. Patent 6,727,618 B1, Apr. 27, 2004. [9] J.T. Marshall, M.E.F. Kasarda, and J. Imlach, “A multipoint measurement technique for the enhancement of force measurement with active magnetic bearings,” ASME Trans., vol. 125, pp. 90−94, Jan. 2003. [10] C.R. Knospe and E.H. Maslen, “Introduction to active magnetic bearing,” presented at NASA Lewis Research Center, Cleveland, OH, Feb. 24−25, 1999.

TABLE I NONLEVITATION MAGNETIC FORCE DATA Coil Upper Coil Lower Coil Derated Current Experimental Experimental Theoretical (A) Radial Force Radial Force Radial Force (N) (N) (N) 0.98 13.21 12.09 12.92 1.04 14.70 13.45 14.46 1.08 16.11 14.76 15.74 1.13 17.61 16.25 17.22 1.18 19.20 17.78 18.77 1.24 20.90 19.54 20.43 1.28 22.66 21.13 22.05 1.33 24.41 22.85 23.76 1.38 26.18 24.68 25.58 1.43 28.11 26.38 27.43 1.49 30.27 28.53 29.57 1.53 32.26 30.54 31.52 1.58 34.33 32.50 33.35 1.63 36.55 34.64 35.45 1.68 38.95 37.08 37.94 1.73 41.28 39.32 40.18 1.78 43.54 41.72 42.39 1.83 46.03 44.18 44.70 1.88 48.77 46.85 47.44 1.93 51.37 49.43 49.99 1.98 53.99 52.00 52.45 2.03 56.64 54.77 55.03 Data taken for 8/6 hybrid bearingless switched-reluctance motor.

TABLE II LEVITATION MAGNETIC FORCE DATA Displacement Derated Experimental Absolute Absolute Absolute (10–5 m) Theoretical Force Force Displacement Theoretical Force Experimental Force (N) (N) (10–5 m) (N) (N) –5.43 –7.51 –8.13 5.43 7.51 8.13 –4.83 –6.96 –7.34 4.83 6.96 7.34 –3.93 –6.08 –6.19 3.93 6.08 6.19 –3.23 –5.03 –5.03 3.23 5.03 5.03 –2.33 –3.70 –3.61 2.33 3.70 3.61 –1.43 –2.37 –2.29 1.43 2.37 2.29 –0.73 –1.20 –1.16 0.73 1.20 1.16 0.00 0.00 0.00 0.00 0.00 0.00 0.81 1.31 1.28 0.81 1.31 1.28 1.52 2.50 2.37 1.52 2.50 2.37 2.51 4.02 3.85 2.51 4.02 3.85 3.19 4.96 4.80 3.19 4.96 4.80 3.94 5.92 5.73 3.94 5.92 5.73 5.05 7.18 7.07 5.05 7.18 7.07 5.76 7.75 7.78 5.76 7.75 7.78 Data taken for 8/6 hybrid bearingless switched-reluctance motor.

Fig. 3. Axial view of the 8/6 hybrid bearingless switched-reluctance motor with rotor and stator pole 1 aligned.

Fig. 1. Disassembled eight-pole stator and six-pole hybrid rotor of bearingless switched-reluctance motor.

Fig. 4. Equivalent two-dimensional rotor-stator flux distribution for nonlevitated 8/6 hybrid bearingless switched-reluctance motor, where for the c circular lamimation φk is magnetic flux between stator pole k and rotor c lamination stack, Ak is cross-sectional area between stator pole k and rotor lamination stack, and g0 is gap width.

Fig. 5. Magnetic equivalent circuit for nonlevitated 8/6 hybrid bearingless c switched-reluctance motor, where for circular lamimation Rk is reluctance c and φk is magnetic flux between stator pole k and rotor lamination stack.

Fig. 2. Side view of circular rotor and stator tooth segment of 8/6 hybrid bearingless switched-reluctance motor, where B is magnetic flux density, g is gap width, ltw is tooth width, r is outer radius of rotor laminations, and v is gap volume.

Fig. 6. Equivalent two-dimensional rotor-stator flux distribution for nonlevitated 8/6 hybrid bearingless switched-reluctance motor, where for s scalloped lamination g0 is gap width, φk is magnetic flux between stator pole s k and rotor lamination stack, and Ak is cross-sectional area between stator pole k and rotor lamination stack.

Fig. 7. Magnetic equivalent circuit for nonlevitated 8/6 hybrid bearingless s switched-reluctance motor, where for scalloped lamination Rk is reluctance s and φk is magnetic flux between stator pole k and rotor lamination stack.

Fig. 11. Magnetic equivalent circuit for levitated 8/6 hybrid bearingless switched-reluctance motor, where ik is current in coil k, Pk is permeance at stator pole k, and φk is magnetic flux at stator pole k.

Fig. 8. Experimental setup used in obtaining nonlevitation magnetic radial forces for 8/6 hybrid bearingless switched reluctance motor (BSRM).

Fig. 12. Experimental setup used in obtaining levitation magnetic radial forces for 8/6 hybrid bearingless switched reluctance motor (BSRM).

Fig. 9. Nonlevitation magnetic radial force curves for 8/6 hybrid bearingless switched-reluctance motor. Experimental least-squares-fit curves (dotted lines) for upper and lower poles and theoretical curve (solid line).

Fig. 13. Experimental (dotted line) and theoretical (solid line) levitation magnetic radial force curves for 8/6 hybrid bearingless switched-reluctance motor.

Fig. 10. Axial view of 8/6 hybrid bearingless switched-reluctance motor with rotor and stator pole 1 unaligned.