A Full-Period Mathematical Model for a Hybrid-Rotor Bearingless Switched Reluctance Motor

: A bearingless switched reluctance motor (BSRM) has the combined characteristics of a switched reluctance motor (SRM) and a magnetic bearing. The hybrid-rotor BSRM (HBSRM) discussed in the paper has a twelve-pole stator and an eight-pole hybrid rotor, which is composed of a cylindrical rotor and a salient-pole rotor. Although the asymmetry of the hybrid rotor makes the structure and magnetic ﬁeld of the HBSRM more complex, it can always produce a signiﬁcant amount of magnetic pulling force to levitate a rotor shaft at all the rotor angular positions of each phase, which is not available in a traditional BSRM. The classical mathematical model for a conventional BSRM is valid only when its rotor rotates from the start of the overlap position to the aligned position, and the radial force and torque derived from this model are discontinuous at the aligned positon, which is harmful to the motor’s stable operation. In this paper, a full-period mathematical model on the assumption that the gap permeance is cut apart by straight lines or improved elliptical lines for a 12/8-pole HBSRM is provided. On the basis of this mathematical model, the continuity of the radial force and torque at all the rotor angular positions can be guaranteed, and the ﬁne characteristics of this mathematical model have been veriﬁed by simulations.


Introduction
Switched reluctance motor (SRM) has superior performance under special environments such as the starter/generator in all electric/multi-electric aeroengine, vacuum pump, and flywheel energy storage systems, because of its simple structure, low cost, robustness, and suitability for operation in high temperatures [1][2][3]. The SRM generally has a short air gap length in order to effectively produce the rotational reluctance torque, and a considerable magnetic attraction force is also simultaneously generated in radial directions. It is very convenient to utilize this radial force so as to levitate a rotor shaft. Therefore, the SRM has a good feature as a bearingless motor, which is characterized by the integration of an electrical motor and magnetic bearing [4][5][6].
A bearingless switched reluctance motor (BSRM) with differential stator-winding configuration was proposed in References [7][8][9][10]. This structure has two kinds of stator windings composed of motor main windings and radial force windings in the same stator in order to produce the required radial force that may realize the suspension of the rotor shaft without mechanical contacts or lubrication. Then, some single-winding BSRMs with the same structures as traditional SRM have also been developed [11][12][13][14]. In these structures, the required radial force and rotational torque can be simultaneously produced by independently controlling currents in all the windings. In a control system of BSRM, fast current regulation is required for radial force and torque control. Especially for continuous magnetic levitation, suspended forces must be as shown in Figure 1. Same as the traditional 12/8-pole SRM, each stator tooth in this motor still has a coil, and the four sets of coils separated by 90° in space form one phase. Therefore, twelve sets of coils are divided into three phases: A-phase, B-phase, and C-phase, and each phase is separated by 30° in space. Three-phase windings are excited in turn every 15°, and the rotor rotates counterclockwise. In the paper, θ indicates the rotor angular position, and θ = 0 is defined as the aligned position of the A-phase. In Figure 2, the rotor angular θ of is −π/24 (−7.5°).  In Figure 2, the blue broken lines show the two-pole fluxes produced by the current ia1 in the winding A1, and the red solid lines show the two-pole fluxes produced by the current ia3 in the winding A3. If ia1 > ia3, the flux density in the air gap a1 is greater than that in the air gap a3. These two unequal magnetic fluxes result in the radial force Fα acting on the rotor shaft toward the positive direction on the α-axis. On the other hand, when ia1 < ia3, the flux in the air gap a1 is less than that in the air gap a3, and then, a negative radial force on the α-axis direction may be generated. Moreover, a radial force Fβ on the β-axis direction can be produced by controlling the currents ia2 in the winding A2 and ia4 in the winding A4. Therefore, the desired magnetic levitation force can be produced in a radial direction. Similarly, this principle of radial force production can be applied to the B-and C-phase. The radial force can be successfully built by these three phases in turn every 15°. Figure 3 shows the distribution of the phase inductance, radial force, and torque in the two types of bearingless motor structures; LHBSRM is the phase inductance of the 12/8- as shown in Figure 1. Same as the traditional 12/8-pole SRM, each stator tooth in this motor still has a coil, and the four sets of coils separated by 90° in space form one phase. Therefore, twelve sets of coils are divided into three phases: A-phase, B-phase, and C-phase, and each phase is separated by 30° in space. Three-phase windings are excited in turn every 15°, and the rotor rotates counterclockwise. In the paper, θ indicates the rotor angular position, and θ = 0 is defined as the aligned position of the A-phase. In Figure 2, the rotor angular θ of is −π/24 (−7.5°).   In Figure 2, the blue broken lines show the two-pole fluxes produced by the current ia1 in the winding A1, and the red solid lines show the two-pole fluxes produced by the current ia3 in the winding A3. If ia1 > ia3, the flux density in the air gap a1 is greater than that in the air gap a3. These two unequal magnetic fluxes result in the radial force Fα acting on the rotor shaft toward the positive direction on the α-axis. On the other hand, when ia1 < ia3, the flux in the air gap a1 is less than that in the air gap a3, and then, a negative radial force on the α-axis direction may be generated. Moreover, a radial force Fβ on the β-axis direction can be produced by controlling the currents ia2 in the winding A2 and ia4 in the winding A4. Therefore, the desired magnetic levitation force can be produced in a radial direction. Similarly, this principle of radial force production can be applied to the B-and C-phase. The radial force can be successfully built by these three phases in turn every 15°. Figure 3 shows the distribution of the phase inductance, radial force, and torque in the two types of bearingless motor structures; LHBSRM is the phase inductance of the 12/8- In Figure 2, the blue broken lines show the two-pole fluxes produced by the current i a1 in the winding A1, and the red solid lines show the two-pole fluxes produced by the current i a3 in the winding A3. If i a1 > i a3 , the flux density in the air gap a1 is greater than that in the air gap a3. These two unequal magnetic fluxes result in the radial force F α acting on the rotor shaft toward the positive direction on the α-axis. On the other hand, when i a1 < i a3 , the flux in the air gap a1 is less than that in the air gap a3, and then, a negative radial force on the α-axis direction may be generated. Moreover, a radial force F β on the β-axis direction can be produced by controlling the currents i a2 in the winding A2 and i a4 in the winding A4. Therefore, the desired magnetic levitation force can be produced in a radial direction. Similarly, this principle of radial force production can be applied to the B-and C-phase. The radial force can be successfully built by these three phases in turn every 15 • . Figure 3 shows the distribution of the phase inductance, radial force, and torque in the two types of bearingless motor structures; L HBSRM is the phase inductance of the 12/8pole HBSRM, and L BSRM is the phase inductance of the 12/8-pole BSRM. In a 12/8-pole BSRM, when its stator tooth and rotor tooth are in a completely misaligned positions or the overlapping area is very small, the rotor cannot suspend, or requires a large current to suspend, the shaft. Thus, there may be a dead zone of levitation in each rotor period of the traditional BSRM, such as regions I and IV, as shown in Figure 3. However, for a 12/8-pole HBSRM, the magnetic pulling force between the stator and cylindrical rotor is enough to suspend the shaft, so the levitation of its rotor shaft can be achieved in all the rotor angular positions. pole HBSRM, and LBSRM is the phase inductance of the 12/8-pole BSRM. In a 12/8-pole BSRM, when its stator tooth and rotor tooth are in a completely misaligned positions or the overlapping area is very small, the rotor cannot suspend, or requires a large current to suspend, the shaft. Thus, there may be a dead zone of levitation in each rotor period of the traditional BSRM, such as regions Ⅰ and Ⅳ, as shown in Figure 3. However, for a 12/8pole HBSRM, the magnetic pulling force between the stator and cylindrical rotor is enough to suspend the shaft, so the levitation of its rotor shaft can be achieved in all the rotor angular positions. In view of the radial force characteristics of the 12/8-pole HBSRM, its excitation range of each phase is redesigned to be the same as that of a conventional SRM, as shown in Figure 4. Taking the A-phase of a 12/8-pole HBSRM as an example, it can be excited from −22.5° to −7.5°. For the A-phase of a 12/8-pole BSRM, its excitation range is generally designed as (−15°, 0). It can be seen from Figure 4, compared with the traditional BSRM, that the switching angle of each phase of the 12/8-pole HBSRM moves forward by 7.5°. This excitation mode has the following two advantages: (a) the current tracking effect at the opening time can be improved due to the smaller change rate of the phase inductance, and (b) the torque ripple can be reduced at a high speed, because the motor running in the negative torque region is avoided. In Figure 4, La, Lb, and Lc are the inductance of the A-, B-, and C-phases of a 12/8-pole HBSRM, and ia1~ia4, ib1~ib4, and ic1~ic4 are the four winding currents of the A-, B-, and C-phases.  In view of the radial force characteristics of the 12/8-pole HBSRM, its excitation range of each phase is redesigned to be the same as that of a conventional SRM, as shown in Figure 4. Taking the A-phase of a 12/8-pole HBSRM as an example, it can be excited from −22.5 • to −7.5 • . For the A-phase of a 12/8-pole BSRM, its excitation range is generally designed as (−15 • , 0). It can be seen from Figure 4, compared with the traditional BSRM, that the switching angle of each phase of the 12/8-pole HBSRM moves forward by 7.5 • . This excitation mode has the following two advantages: (a) the current tracking effect at the opening time can be improved due to the smaller change rate of the phase inductance, and (b) the torque ripple can be reduced at a high speed, because the motor running in the negative torque region is avoided. In Figure 4, L a , L b , and L c are the inductance of the A-, B-, and C-phases of a 12/8-pole HBSRM, and i a1~ia4 , i b1~ib4 , and i c1~ic4 are the four winding currents of the A-, B-, and C-phases. pole HBSRM, and LBSRM is the phase inductance of the 12/8-pole BSRM. In a 12/8-pole BSRM, when its stator tooth and rotor tooth are in a completely misaligned positions or the overlapping area is very small, the rotor cannot suspend, or requires a large current to suspend, the shaft. Thus, there may be a dead zone of levitation in each rotor period of the traditional BSRM, such as regions Ⅰ and Ⅳ, as shown in Figure 3. However, for a 12/8pole HBSRM, the magnetic pulling force between the stator and cylindrical rotor is enough to suspend the shaft, so the levitation of its rotor shaft can be achieved in all the rotor angular positions. In view of the radial force characteristics of the 12/8-pole HBSRM, its excitation range of each phase is redesigned to be the same as that of a conventional SRM, as shown in Figure 4. Taking the A-phase of a 12/8-pole HBSRM as an example, it can be excited from −22.5° to −7.5°. For the A-phase of a 12/8-pole BSRM, its excitation range is generally designed as (−15°, 0). It can be seen from Figure 4, compared with the traditional BSRM, that the switching angle of each phase of the 12/8-pole HBSRM moves forward by 7.5°. This excitation mode has the following two advantages: (a) the current tracking effect at the opening time can be improved due to the smaller change rate of the phase inductance, and (b) the torque ripple can be reduced at a high speed, because the motor running in the negative torque region is avoided. In Figure 4, La, Lb, and Lc are the inductance of the A-, B-, and C-phases of a 12/8-pole HBSRM, and ia1~ia4, ib1~ib4, and ic1~ic4 are the four winding currents of the A-, B-, and C-phases.

Mathematical Model
For the HBSRM discussed in the paper, the asymmetry of the hybrid-rotor structure makes its magnetic field be distributed asymmetrically. In order to simplify the magnetic circuit analysis and deduce the mathematical model conveniently, the 12/8-pole HBSRM is equivalent to two parts, namely a 12/8-pole BSRM and a 12-pole AMB. Moreover, the three-dimensional magnetic field of HBSRM can also be regarded as two two-dimensional fields. The magnetic fluxes of two equivalent parts produced by the A-phase current i a1~ia4 are as shown in Figure 5. The dimensions of a prototype are shown in Table 1.

Mathematical Model
For the HBSRM discussed in the paper, the asymmetry of the hybrid-rotor structure makes its magnetic field be distributed asymmetrically. In order to simplify the magnetic circuit analysis and deduce the mathematical model conveniently, the 12/8-pole HBSRM is equivalent to two parts, namely a 12/8-pole BSRM and a 12-pole AMB. Moreover, the three-dimensional magnetic field of HBSRM can also be regarded as two two-dimensional fields. The magnetic fluxes of two equivalent parts produced by the A-phase current ia1~ia4 are as shown in Figure 5. The dimensions of a prototype are shown in Table 1.   Figure 6 shows the magnetic equivalent circuit of the 12/8-pole HBSRM. The voltage sources indicate magnetomotive forces of these twelve windings, and the air gap permeances are expressed as equivalent resistances. The magnetic equivalent circuit of the Aphase is shown in Figure 7, where N represents the number of turns of each winding in each stator tooth, Pa1~Pa4 show the air gap permeances of the A-phase stator teeth, and Φa1~Φa4 indicate the magnetic fluxes of each stator tooth. Only the magnetic equivalent circuit of the A-phase is used to make a simple calculation because of a little mutual inductance among the A-, B-, and C-phases in a traditional SRM.   Figure 6 shows the magnetic equivalent circuit of the 12/8-pole HBSRM. The voltage sources indicate magnetomotive forces of these twelve windings, and the air gap permeances are expressed as equivalent resistances. The magnetic equivalent circuit of the A-phase is shown in Figure 7, where N represents the number of turns of each winding in each stator tooth, P a1~Pa4 show the air gap permeances of the A-phase stator teeth, and Φ a1~Φa4 indicate the magnetic fluxes of each stator tooth. Only the magnetic equivalent circuit of the A-phase is used to make a simple calculation because of a little mutual inductance among the A-, B-, and C-phases in a traditional SRM.  If each permeance shown as Pa1~Pa4 can simply be assumed, the self and mutual inductances of each winding of the A-phase can be calculated from Figure 7. The detailed derivations are similar to that of the 12/8-pole double-winding BSRM in References [7,8]. The self-inductances and the mutual inductances are expressed as

Derivation of Inductances
where La1~La4 are the self-inductances of the windings A1~A4, Ma12 is the mutual inductance between A1 and A2, Ma13 is the mutual inductance between A1 and A3, Ma14 is the mutual inductance between A1 and A4, Ma23 is the mutual inductance between A2 and A3, Ma24 is the mutual inductance between A2 and A4, Ma34 is the mutual inductance between A3 and A4, and P is the sum of the permeances Pa1~Pa4. As shown in Figure 8, the permeance Pa1 of the air gap a1 is divided into three parts of permeances: P1~P3. In this paper, P1 is derived by using a straight magnetic path, and P2 and P3 are calculated by the assumed magnetic paths. In Figure 8, t represents an angular position on a rotor circular surface, dt is a derivative of t, lg is the air gap length after the rotor eccentricity, l0 is the average air gap length without the rotor eccentricity, α is the  If each permeance shown as Pa1~Pa4 can simply be assumed, the self and mutual inductances of each winding of the A-phase can be calculated from Figure 7. The detailed derivations are similar to that of the 12/8-pole double-winding BSRM in References [7,8]. The self-inductances and the mutual inductances are expressed as

Radial Forces in the BSRM Part
where La1~La4 are the self-inductances of the windings A1~A4, Ma12 is the mutual inductance between A1 and A2, Ma13 is the mutual inductance between A1 and A3, Ma14 is the mutual inductance between A1 and A4, Ma23 is the mutual inductance between A2 and A3, Ma24 is the mutual inductance between A2 and A4, Ma34 is the mutual inductance between A3 and A4, and P is the sum of the permeances Pa1~Pa4. As shown in Figure 8, the permeance Pa1 of the air gap a1 is divided into three parts of permeances: P1~P3. In this paper, P1 is derived by using a straight magnetic path, and P2 and P3 are calculated by the assumed magnetic paths. In Figure 8, t represents an angular position on a rotor circular surface, dt is a derivative of t, lg is the air gap length after the rotor eccentricity, l0 is the average air gap length without the rotor eccentricity, α is the If each permeance shown as P a1~Pa4 can simply be assumed, the self and mutual inductances of each winding of the A-phase can be calculated from Figure 7. The detailed derivations are similar to that of the 12/8-pole double-winding BSRM in References [7,8]. The self-inductances and the mutual inductances are expressed as

Radial Forces in the BSRM Part
P P a1 (P a2 + P a3 + P a4 ), L a2 = N 2 P P a2 (P a1 + P a3 + P a4 ), L a3 = N 2 P P a3 (P a1 + P a2 + P a4 ), L a4 = N 2 P P a4 (P a1 + P a2 + P a3 ), M a12 = N 2 P P a1 P a2 , M a13 = − N 2 P P a1 P a3 , M a14 = N 2 P P a1 P a4 , M a23 = N 2 P P a2 P a3 , M a24 = − N 2 P P a2 P a4 , M a34 = N 2 P P a3 P a4 (1) where L a1~La4 are the self-inductances of the windings A1~A4, M a12 is the mutual inductance between A1 and A2, M a13 is the mutual inductance between A1 and A3, M a14 is the mutual inductance between A1 and A4, M a23 is the mutual inductance between A2 and A3, M a24 is the mutual inductance between A2 and A4, M a34 is the mutual inductance between A3 and A4, and P is the sum of the permeances P a1~Pa4 . As shown in Figure 8, the permeance P a1 of the air gap a1 is divided into three parts of permeances: P 1~P3 . In this paper, P 1 is derived by using a straight magnetic path, and P 2 and P 3 are calculated by the assumed magnetic paths. In Figure 8, t represents an angular position on a rotor circular surface, dt is a derivative of t, l g is the air gap length after the rotor eccentricity, l 0 is the average air gap length without the rotor eccentricity, α is the radial displacement of the rotor in the α-axis direction, r is the rotor radius, and dP 2 and dP 3 are the permeances of the infinitesimal width of the assumed magnetic path. radial displacement of the rotor in the α-axis direction, r is the rotor radius, and dP2 and dP3 are the permeances of the infinitesimal width of the assumed magnetic path. The assumed magnetic paths are composed of improved elliptical lines using a variable k. The finite element analysis of an electromagnetic field in the literature [16] showed that it relies on the rotor angular position θ, the rotor radius r, the average air gap length l0, and the variable a. Reference [16] used the least squares method to calculate the variable a from the finite element analysis. In the paper, in order to simplify the derivation and computation, the coefficient k is directly simplified as

Radial Forces in the BSRM
According to the definition of permeance, dP2 can be written as where μ0 is permeability in the air, s represents the cross-sectional area of dP2, ds is a derivative of s, and l is the average length of the magnetic path. As shown in Figure 8, the short half-axis of the elliptical line is t, and the long halfaxis may be written as l0 + kt. Correspondingly, the average cross-sectional area ds of dP2 can be approximately expressed as where hs is the stack length of the scalloped lamination segments. The average length of the assumed magnetic path is approximately expressed as Substituting Equations (2), (4), and (5) into Equation (3) yields Moreover, the permeance P2 can be calculated as ( ) Similarly, the permeance P3 is derived considering the symmetrical fringing flux paths as The assumed magnetic paths are composed of improved elliptical lines using a variable k. The finite element analysis of an electromagnetic field in the literature [16] showed that it relies on the rotor angular position θ, the rotor radius r, the average air gap length l 0 , and the variable a. Reference [16] used the least squares method to calculate the variable a from the finite element analysis. In the paper, in order to simplify the derivation and computation, the coefficient k is directly simplified as k = |θ|r l 0 + |θ|r (2) According to the definition of permeance, dP 2 can be written as where µ 0 is permeability in the air, s represents the cross-sectional area of dP 2 , ds is a derivative of s, and l is the average length of the magnetic path. As shown in Figure 8, the short half-axis of the elliptical line is t, and the long half-axis may be written as l 0 + kt. Correspondingly, the average cross-sectional area ds of dP 2 can be approximately expressed as where h s is the stack length of the scalloped lamination segments. The average length of the assumed magnetic path is approximately expressed as Substituting Equations (2), (4), and (5) into Equation (3) yields Moreover, the permeance P 2 can be calculated as Similarly, the permeance P 3 is derived considering the symmetrical fringing flux paths as The permeance P 1 of the air gap between the rotor and stator poles is expressed as Moreover, the total permeances P a1 can be obtained by summing P 1 , P 2 , and P 3 as Similarly, P a2 , P a3 , and P a4 at the positions of 0 ≤ |θ| ≤ π 12 can be derived by using the above method.
(b) π 12 ≤ |θ| ≤ π 8 Figure 9 shows the enlarged magnetic flux of the BSRM part at the unaligned position obtained by a two-dimensional finite element analysis. The permeance P a1 of the air gap a1 is composed of four parts of permeances: P 4~P7 . Similar to the assumed magnetic path at the positions of 0 ≤ |θ| ≤ π 12 , as shown in Figure 8, the shape of the fringing magnetic paths is still set in the shape of an ellipse. When π 12 ≤ |θ| ≤ π 8 , the assumed magnetic paths of the air gap a1 in the BSRM part are shown in Figure 10. dP 4 and dP 5 are the permeances of the infinitesimal width of the assumed magnetic path.
Similarly, Pa2, Pa3, and Pa4 at the positions of 0 12 π θ ≤ ≤ can be derived by using th above method.
(b) 12 8 π π θ ≤ ≤ Figure 9 shows the enlarged magnetic flux of the BSRM part at the unaligned position obtained by a two-dimensional finite element analysis. The permeance Pa1 of the air gap a1 is composed of four parts of permeances: P4~P7. Similar to the assumed magnetic path at the positions of 0 12 π θ ≤ ≤ , as shown in Figure 8, the shape of the fringing magneti paths is still set in the shape of an ellipse. When 12 8 π π θ ≤ ≤ , the assumed magnetic path of the air gap a1 in the BSRM part are shown in Figure 10. dP4 and dP5 are the permeance of the infinitesimal width of the assumed magnetic path.
Similarly, Pa2, Pa3, and Pa4 at the positions of 0 12 π θ ≤ ≤ can be derived by using the above method. π θ ≤ ≤ , as shown in Figure 8, the shape of the fringing magnetic paths is still set in the shape of an ellipse. When 12 8 π π θ ≤ ≤ , the assumed magnetic paths of the air gap a1 in the BSRM part are shown in Figure 10. dP4 and dP5 are the permeances of the infinitesimal width of the assumed magnetic path.  Considering the symmetry of the fringing magnetic paths, the total permeances P a1 at the positions of π 12 ≤ |θ| ≤ π 8 can be written as For the part of permeance P 4 , we can get the following expressions: )r dP 4 = µ 0 ×ds 1 l 1 (12) where l 1 is the average length of the equivalent magnetic path, s 1 is the cross-sectional area of dP 4 , ds 1 is a derivative of s 1 , and the variable k 1 is used to determine the shape of the ellipse at the positions of π 12 ≤ |θ| ≤ π 8 . Moreover, the permeance P 4 may be obtained as Similarly, for the part of permeance P 5 , some related expressions can be written as where l 2 is the average length of the assumed magnetic path, s 2 is the cross-sectional area of dP 5 , ds 2 is a derivative of s 2 , and the variable k 2 is used for calculating the shape of the ellipse at the positions of π 12 ≤ |θ| ≤ π 8 . Then, the total permeance P 5 may be expressed as Substituting Equations (13) and (15) into Equation (11) yields P a1 = 2µ 0 h s π l 0 + 2 |θ| − π 12 r l 0 + |θ| − π 12 r ln 2l 2 0 + π 2 12 r(l 0 + α) 2l 2 0 + πr |θ| − π 12 (l 0 + α) Similarly, P a2 , P a3 , and P a4 at the positions of π 12 ≤ |θ| ≤ π 8 can also be derived following the above method.

Derivation of Radial Forces in BSRM Part
Substituting each permeance calculated from the above method into Equation (1) The A-phase-stored magnetic energy W a in the BSRM part of a HBSRM can be written using Equation (17) as Since the radial force is dependent on the derivative of the stored magnetic energy with respect to the radial displacement of the shaft, the radial forces F α1 and F β1 generated by the A-phase in the BSRM part can be written as where β is the radial displacement of the rotor in the β-axis direction, and K f1 is the coefficient of the radial forces in the BSRM part. It can be expressed as

Radial Forces in the AMB Part
The permeance P a1 of the air gap a1 in the AMB part consists of straight magnetic paths, as shown in Figure 11, and it can be derived as where h c is the stack length of the circular lamination segments. Similarly, P a2 , P a3 , and P a4 in the AMB part can also be obtained by the above method.

Radial Forces in the AMB Part
The permeance Pa1 of the air gap a1 in the AMB part consists of stra paths, as shown in Figure 11, and it can be derived as Where hc is the stack length of the circular lamination segments. Similarly, in the AMB part can also be obtained by the above method. Similarly, the radial forces Fα2 and Fβ2 resulted by the A-phase in the be computed from the derivatives of the stored magnetic energy with respe ments α and β, respectively. The radial forces in the AMB part can be writt where Kf2 is the coefficient of the radial forces in the AMB part. It can be ex Similarly, the radial forces F α2 and F β2 resulted by the A-phase in the AMB part can be computed from the derivatives of the stored magnetic energy with respect to displacements α and β, respectively. The radial forces in the AMB part can be written as where K f2 is the coefficient of the radial forces in the AMB part. It can be expressed as

Total Radial Forces of the HBSRM
The total radial forces of the HBSRM are the sum of both the AMB and BSRM parts. Based on Equations (19), (20), (22), and (23), the total radial forces can be written as where K f is the total coefficient of the radial forces. It can be expressed as

Mathematical Model of the Double-Winding Structure
In essence, a BSRM uses the asymmetry of air gap magnetic fluxes in the radial direction to generate suspended forces. These asymmetric air gap magnetic fluxes can be generated by single-winding or double-winding excitations. Figure 12 shows the A-phase winding configuration and magnetic flux lines by finite element analysis in a 12/8-pole double-winding BSRM. The motor main winding N ma consists of four coils connected in a series. On the other hand, the radial force windings N sa1 and N sa1 consist of two coils each. The B-phase winding configuration is situated on the one-third rotational position of the A-phase. The C-phase winding configuration is situated on the two-thirds rotational position of the A-phase.

Radial force winding Nsa2
Radial force winding Nsa1 Motor main winding N ma Based on the principle of magnetomotive force balance, the equivalent relations between single-winding and double-winding structures can be expressed as where Nm is the number of turns of the motor main winding, Nb is the number of turns of radial force winding, ima is the current in the motor main winding Nma, isa1 is the current in the radial force winding Nsa1, and isa2 is the current in the radial force winding Nsa1.
Substituting Equation (28) into Equations (24) and (26), the radial force and torque expressions of the HBSRM with double-winding configurations on each stator tooth can be written as It can be seen from Equations (29) and (30) that the expressions are consistent with those in References [7,8]. Therefore, the radial force and torque coefficients derived from the single-winding structure can be directly applied to the double-winding BSRM. Figure 13 shows the radial force and torque coefficients obtained by the classical model and novel model of the BSRM part based on the parameters in Table 1. It is known from Figure 13 that the torque coefficient derived from the classical method is not zero at the position of θ = 0, and the radial force coefficient obtained by the classical method is discontinuous at the aligned position. However, these problems can be solved by the new model presented in this paper, because the torque is proportional to the derivative of the inductance with respect to the rotor angular position θ. Moreover, if the derivative of the gap permeance at the aligned position is continuous, the torque is also continuous at this position. The continuity of the new model is analyzed as follows: Based on the principle of magnetomotive force balance, the equivalent relations between single-winding and double-winding structures can be expressed as

Model Analysis
where N m is the number of turns of the motor main winding, N b is the number of turns of radial force winding, i ma is the current in the motor main winding N ma , i sa1 is the current in the radial force winding N sa1 , and i sa2 is the current in the radial force winding N sa1 .
Substituting Equation (28) into Equations (24) and (26), the radial force and torque expressions of the HBSRM with double-winding configurations on each stator tooth can be written as It can be seen from Equations (29) and (30) that the expressions are consistent with those in References [7,8]. Therefore, the radial force and torque coefficients derived from the single-winding structure can be directly applied to the double-winding BSRM. Figure 13 shows the radial force and torque coefficients obtained by the classical model and novel model of the BSRM part based on the parameters in Table 1. It is known from Figure 13 that the torque coefficient derived from the classical method is not zero at the position of θ = 0, and the radial force coefficient obtained by the classical method is discontinuous at the aligned position. However, these problems can be solved by the new model presented in this paper, because the torque is proportional to the derivative of the inductance with respect to the rotor angular position θ. Moreover, if the derivative of the gap permeance at the aligned position is continuous, the torque is also continuous at this position. The continuity of the new model is analyzed as follows:

Model Analysis
In addition, the torque coefficient at the positions of θ = ±π/12 may be expressed as The novel model The classical model in reference [7] θ/(°) When the rotor rotates from the positive torque period of θ < 0 to the aligned position, the derivative of P a1 at the aligned position can be expressed as Similarly, if the rotor oppositely rotates from the negative torque period of θ > 0 to the aligned position, the derivative of P a1 at the aligned position is written as Moreover, the derivative of P a1 at the unaligned positions of θ = ±π/8 can be written as In addition, the torque coefficient at the positions of θ = ±π/12 may be expressed as In an SRM, the torques at the aligned and unaligned positions invariably become zero. It can be seen from Equations (31)-(33) that the torque expressions proposed in the paper at the aligned and unaligned positions are zero, and the torques at these positions are continuous. Otherwise, it is noted from Equation (34) that the novel model also ensures the consistency of the torque, although it is derived from different regions and assumed magnetic paths. Moreover, the continuity of the radial force formulas at all the rotor angular positons can also be verified using a similar method. Figure 14 shows the calculated results of the radial forces and torques using the parameters of the prototype shown in Table 1. A two-dimensional finite element analysis for the A-phase of HBSRM was carried out in this paper. In the radial force calculations, the currents in four windings of the A-phase are, respectively, set as follows: i a1 = 4A, i a2 = 2A, i a3 = 0, and i a4 = 2A. In the torque simulation, the A-phase current is set to i a1 = i a2 = i a3 = i a4 = 4A. It can be seen from Figure 14 that the novel full-period mathematical model has good accuracy, and both the radial force and torque are continuous at all the positions of each rotor period.
sures the consistency of the torque, although it is derived from different regions and assumed magnetic paths. Moreover, the continuity of the radial force formulas at all the rotor angular positons can also be verified using a similar method. Figure 14 shows the calculated results of the radial forces and torques using the parameters of the prototype shown in Table 1. A two-dimensional finite element analysis for the A-phase of HBSRM was carried out in this paper. In the radial force calculations, the currents in four windings of the A-phase are, respectively, set as follows: ia1 = 4A, ia2 = 2A, ia3 = 0, and ia4 = 2A. In the torque simulation, the A-phase current is set to ia1 = ia2 = ia3 = ia4 = 4A. It can be seen from Figure 14 that the novel full-period mathematical model has good accuracy, and both the radial force and torque are continuous at all the positions of each rotor period. The radial force results in Figure 14 further show that the HBSRM has the ability to suspend the shaft in any rotor angular position of each phase. Moreover, this structure can solve the problem that is restricting the performance improvement of the traditional BSRM due to conflicts between the effective output regions of the torque and radial force. Therefore, based on this new model presented in this paper, the radial force excitation interval of each phase and more flexible control algorithm for the HBSRM can be further optimized to improve the motor's output performance.

Conclusions
A 12/8-pole BSRM with a hybrid rotor composed of circular and scalloped lamination segments was discussed in this paper. The cylindrical rotor segment of the HBSRM was used to generate additional radial forces, which could solve the problem that the radial force of a traditional BSRM near the unaligned positions of each phase is insufficient to levitate a rotor shaft. In order to simplify the asymmetric magnetic field of the 12/8-pole HBSRM, this motor was regarded as a 12/8-pole BSRM and a 12-pole AMB, and then, the equivalent magnetic circuits of both parts were established for the derivation of the radial force and torque formulas. Moreover, a novel full-period mathematical model for the 12/8pole HBSRM was derived, which was applicable to any rotor angular position of each phase. Compared with the classical model of the traditional BSRM, the new model can ensure the continuity of the radial force and torque at all the rotor positions, and the excellent characteristics of this model were confirmed by a finite element analysis. In addition, the full-period model for the HBSRM was quite helpful in optimizing this motor's The radial force results in Figure 14 further show that the HBSRM has the ability to suspend the shaft in any rotor angular position of each phase. Moreover, this structure can solve the problem that is restricting the performance improvement of the traditional BSRM due to conflicts between the effective output regions of the torque and radial force. Therefore, based on this new model presented in this paper, the radial force excitation interval of each phase and more flexible control algorithm for the HBSRM can be further optimized to improve the motor's output performance.

Conclusions
A 12/8-pole BSRM with a hybrid rotor composed of circular and scalloped lamination segments was discussed in this paper. The cylindrical rotor segment of the HBSRM was used to generate additional radial forces, which could solve the problem that the radial force of a traditional BSRM near the unaligned positions of each phase is insufficient to levitate a rotor shaft. In order to simplify the asymmetric magnetic field of the 12/8-pole HBSRM, this motor was regarded as a 12/8-pole BSRM and a 12-pole AMB, and then, the equivalent magnetic circuits of both parts were established for the derivation of the radial force and torque formulas. Moreover, a novel full-period mathematical model for the 12/8-pole HBSRM was derived, which was applicable to any rotor angular position of each phase. Compared with the classical model of the traditional BSRM, the new model can ensure the continuity of the radial force and torque at all the rotor positions, and the excellent characteristics of this model were confirmed by a finite element analysis. In addition, the full-period model for the HBSRM was quite helpful in optimizing this motor's radial force excitation interval of each phase and the control algorithm for the improvement of the output performance.