Research on the Bearingless Brushless DC Motor Structure with Like-Tangential Parallel-Magnetization Interpolar Magnetic Poles and Its Air-Gap Magnetic Field Analytical Calculation

Abstract

This work focuses on the small Bearingless Brushless DC Motor (BL-BLDCM), to solve the problems, such as larger commutation torque ripple and difficult solution of air-gap magnetic field, a novel BL-BLDCM structure with like-tangential parallel-magnetization interpolar magnetic poles (LTPMIMPs) is proposed, which is abbreviated as BL-BLDCM-LTPMIMP in this work, and the analytical calculation model of its air-gap magnetic field has been investigated. First, inserting a like-tangential parallel magnetizing auxiliary magnetic pole between every two adjacent single-radial-magnetizing main poles, and forming several combination magnetic poles, each of which is composed of a radial-magnetizing main magnetic pole and two semi-auxiliary-magnetic-poles (with different magnetization directions) located on both sides. Then, by solving the Laplace equation and Poisson equation in every subdomain, and combining the relative permeability function, the analytical expressions of the air-gap magnetic fields for the BL-BLDCM-LTPMIMP was obtained. The armature reaction magnetic fields of the torque windings and suspension windings are also analyzed. Finally, through the finite element method (FEM), the correctness and computational accuracy of the analytical calculation model for the air-gap magnetic field is proven. Additionally, the comparison of electromagnetic characteristics with ordinary BL-BLDCM shows that the BL-BLDCM-LTPMIMP can not only effectively improve the amplitude and stability of electromagnetic torque on the basis of obtaining a shoulder-shrugged trapezoidal wave air-gap magnetic field but also has stable radial magnetic levitation force control characteristics.

1. Introduction

Although magnetic bearings have been widely used for rotor support in high-speed motors [1], they still have the disadvantages of high-power consumption for magnetic levitation control, long rotor shafts, and limited critical speed, etc. [2]. Bearingless motor is a new type of magnetic levitation motor developed based on the structural similarity between magnetic bearings and AC motor stator [2,3]. Due to its advantages of no mechanical abrasion, no pollution, and being suitable for ultra-high-speed operation, it has broad application prospects in aerospace, electric vehicles, electromechanical flywheel energy storage, artificial heart pumps, life sciences, and other fields [2,4,5,6]. Bearingless motor technology has been applied in induction motors [3,4], switched reluctance motors [7], synchronous reluctance motor [5], and permanent magnet synchronous motors [8,9,10]. Among them, Bearingless Brushless DC Motor (BL-BLDCM) has gradually become an international research hotspot due to inheriting the high efficiency, energy saving, large starting torque, and high-power density characteristics of the ordinary brushless DC motor [2,11,12,13]. To generate a trapezoidal wave air-gap magnetic field, BL-BLDCM usually adopts radial magnetization method for its rotor permanent magnets [11,13]. However, when this type of magnetization is adopted, there is a problem of air-gap magnetic field weakening at the two ends of the flat-top waveform, which is one of the causes of BL-BLDCM commutation torque pulsation.

The air-gap magnetic field distribution has an important influence on motor performance, and establishing an accurate mathematical model is the key to solving motor magnetic field problems. Compared with the finite element method (FEM), the analytical calculation method has stronger timeliness and universality in obtaining the change rule of relevant physical quantities with the motor structure parameters [14,15]. Regarding ordinary brushless DC motors, there have been in-depth studies internationally on analytical modeling [16,17,18,19], as well as the optimization of permanent magnet structures [17,18,19]. Reference [15] proposed an analytical modeling method for surface-mounted permanent magnet brushless DC motor and, by means of the angle-preserving transformation and the 2 d relative permeability function, analyzed the effect of stator slotting on the magnetic field distribution within the air-gap and magnet regions and analyzed the distributions of open circuit magnetic field and armature reactive magnetic field under load condition. Reference [17] proposed a radial and parallel combination magnetization method for the brushless DC motor. The unloaded air-gap magnetic field was also calculated, and the influence laws of motor parameters such as pole arc coefficient, remanent magnetization rate, and permanent magnet thickness on the air-gap magnetic field distribution were derived. In order to improve the electromagnetic torque, references [18,19] applied Halbach arrays to the brushless DC motor, and performed analytical calculations for air-gap magnetic field and electromagnetic torque. For permanent magnet synchronous motors with quasi-Halbach arrays, reference [20] introduced equivalent permanent magnet currents instead of radial or parallel magnetized Halbach arrays, based on which a magnetic field separation theory for magnetic field prediction was proposed. Reference [21] investigated the characteristics of tubular-linear synchronous quasi-Halbach machines, where radially magnetized permanent magnets are replaced by four parallel magnetized permanent magnet arrays for cost-effectiveness. And the air-gap flux density was calculated analytically. Reference [22] combined the permanent magnet equivalent surface current method and the slotted sub-area model to derive an analytical equation for the air-gap magnetic field of the brushless DC motor considering the impact of slotting. In reference [23], based on the two-dimensional analytical model of a brushless DC motor, an analytical model for calculating the effect of pole shift on the output parameters of the motor is proposed; the effects of pole shift on the radial and tangential components of magnetization vector, as well as on the cogging torque pulsation and counter potential waveform, are also investigated.

For bearingless motor, preliminary research has been conducted internationally on analytical modeling and structural optimization, while current research mainly focuses on bearingless permanent magnet synchronous motors. Reference [14] proposed an improved subdomain modeling method for analytically calculating the magnetic field of a bearingless flux-switched permanent magnet motor by equivalently analyzing the currents of distributed double-layer winding. Reference [24] considered the influences of permanent magnet built-in and stator slotting on the air-gap magnetic field, established a global analytical model of the no-load air gap magnetic field of a bearingless alternating-pole permanent magnet motor, and solved the air-gap magnetic field by the two-dimensional analytical method. Reference [25] adopted a subdomain modeling approach to solve the analytical magnetic field of a slotless bearingless flux-switched permanent magnet motor and proposed a doubly salient relative permeance method to predict the magnetic field of the motor. Reference [26] developed an analytical model of a bearingless permanent magnet synchronous motor based on the subdomain modeling method and used a multi-objective gray-wolf algorithm to optimize the design of the motor structure. At present, the mathematical modeling of the BL-BLDCM is mainly based on the equivalent magnetic circuit method [13,27], and there is still few research on the magnetic pole structure optimization and on the modeling using air-gap magnetic field analysis method.

In this work, to meet the demand for the single-winding bearingless motor technology in the low-power field, on the basis of the ordinary 12-slot 4-pole BL-BLDCM structure, like-tangential parallel-magnetization auxiliary permanent magnetic poles (LTPMIMPs) are added between the radially magnetized main permanent magnetic poles, and then constitute a novel BL-BLDCM structure with LTPMIMPs, which is abbreviated as BL-BLDCM-LTPMIMP in this work. Then, based on the two-dimensional analytical method and the magnetic field boundary conditions, the exact analytical calculation equation for air-gap magnetic flux density of a slotless BL-BLDCM-LTPMIMP is derived, and by introducing the relative permeability function, the air-gap magnetic flux density expression for the slotted BL-BLDCM-LTPMIMP is solved; at the same time, the analytical calculation model of the armature reaction magnetic field of the torque windings and suspension windings are derived. Finally, the analytical computation model of the magnetic field for a BL-BLDCM-LTPMIMP is verified by the FEM, and the performance is analyzed in comparison with that of the ordinary BL-BLDCM.

2. Structure and Working Principle of BL-BLDCM-LTPMIMP

2.1. Structure of BL-BLDCM-LTPMIP

Figure 1 is the schematic structure of the 12-slot 4-pole BL-BLDCM-LTPMIMP investigated in this work. A surface-mounted permanent magnet inner rotor structure is adopted; the magnet poles consist of radially magnetized main poles and (like-tangential parallel-magnetization) interpole auxiliary poles (cf. Figure 2). There are two sets of centralized windings placed in the stator slots, including an outer torque winding and an inner suspension winding. The specific winding structure is shown below.

(1) The torque winding consists of A, B, and C three-phase windings that symmetrically distributing in space, in which A1, A2, A3, and A4 are connected in reverse series, in turn, to form the A-phase torque winding; B1, B2, B3, and B4 are connected in reverse series, in turn, to form the B-phase torque winding; C1, C2, C3, and C4 are connected in reverse series, in turn, to form the C-phase torque winding.

(2) The suspension winding consists of six sets of independent windings, in which the u1 coil is connected in reverse series with the u3 coil to form the U1 winding, the u2 coil is connected in reverse series with the u4 coil to form the U2 winding, and the U1 and U2 form the U-phase suspension winding. Similarly, the v1 coil is connected in reverse series with the v3 coil to form the V1 winding, the v2 coil is connected in reverse series with the v4 coil to form the V2 winding, and the V1 and V2 form the V-phase suspension winding. The w1 coil is connected in reverse series with the w3 coil to form the W1 winding, the w2 coil is connected in reverse series with the w4 coil to form the W2 winding, and W1 and W2 form the W-phase suspension winding.

(3) The three-phase torque windings are connected in a star configuration. The three-phase suspension windings consist of six separated windings.

In this work, auxiliary magnetic poles are added between two adjacent radially magnetized main poles in order to obtain a shoulder-shrugged trapezoidal wave air-gap magnetic field and improve the commutation torque characteristics of a BL-BLDCM. The interpolar auxiliary permanent magnet adopts the like-tangential parallel-magnetization method in this work. Specifically, the geometric centerline of the auxiliary pole is used as the benchmark, and along its perpendicular direction, the like-tangential parallel magnetization direction is to be determined. After two auxiliary magnetic poles are parallel magnetized in the counterclockwise direction, and the other two auxiliary magnetic poles are parallel magnetized in the clockwise direction, a total of two counterclockwise and two clockwise like-tangential parallel-magnetization interpolar poles are achieved.

The schematic diagram of the magnetization directions of the main magnetic poles and the interpole auxiliary magnetic poles are shown in Figure 2, where the pole-arc coefficient of the main magnetic pole is α and the pole-arc coefficient of the auxiliary magnetic pole is denoted as “1 − α”. Each combined magnetic pole is composed of a main magnetic pole and two semi-auxiliary-magnetic-poles located on both sides (with different magnetization directions).

2.2. Working Principle of BL-BLDCM-LTPMIP

Setting the counterclockwise direction as the positive rotation direction, the torque control of the BL-BLDCM-LTPMIMP adopts the “three-phase, two-conduction, six-state” control mode of an ordinary brushless DC motor [27]; the details of how it works are not described here.

According to the different position areas of the rotor, each single-phase suspension winding is used sequentially to control the radial magnetic levitation force [27]. To reduce the coupling degree between the torque system and magnetic levitation system, the torque winding and suspension winding wound around the same stator teeth do not conduct at the same time. In this work, the generation of suspension force along the horizontal X-axis is taken as an example, and the suspension force generation principle can be explained as follows:

(1) At the zero moment, when the excitation currents shown in Figure 1 are injected into the U1 and U2 windings, the magnetic field in the region near the u1 and u4 coils is enhanced, and the magnetic field in the region near the u2 and u3 coils is weakened. According to the principle of electromagnetism, a radial electromagnetic force will be generated at this point along the direction of magnetic field enhancement (i.e., along the direction of the teeth corresponding to the u1 and u4 coils).

(2) By adjusting the current magnitude of the U1 and U2 windings, the magnitude and phase of the resultant radial electromagnetic force (i.e., the synthetic radial magnetic levitation force acting on the rotor) can be changed. When the currents in the U1 and U2 windings satisfy the conditions of “i_U1 > 0, i_U2 = −$\sqrt{3}$i_U1”, a resultant radial magnetic levitation force along the positive direction of the X-axis can be generated.

(3) When radial magnetic levitation force is generated by the V-phase or W-phase suspension winding, the principle is similar.

3. Analytical Calculation of the Air-Gap Magnetic Field of BL-BLDCM-LTPMIMP

3.1. Analysis of Magnetization Intensity

In order to improves the control characteristics of BL-BLDCM body, the BL-BLDCM-LTPMIMP in this work inserts like-tangential parallel-magnetization interpolar auxiliary magnetic poles between the radial-magnetization main magnetic poles of an ordinary BL-BLDCM structure. In order to facilitate the comparison with the ordinary BL-BLDCM structure (see later Section 4), when performing analytical calculations of the air-gap magnetic field, except for the structure parameters of the permanent magnet combination magnetic poles, other structure parameters of BL-BLDCM-LTPMIMP, such as the dimensions of stator and rotor cores and the coil’s turn number, etc., adopt the relevant parameters of an ordinary BL-BLDCM [27]. The BL-BLDCM-LTPMIMP studied in this work is rated at 1500 r/min.

Table 1. Structural parameters of a BL-BLDCM-LTPMIMP.

Parameter	Value	Parameter	Value
Outer radius of stator/(mm)	50	Permanent magnet material	NdFe35
Inner radius of stator/(mm)	25	Main magnetic pole arc coefficient α	0.9
Rotor radius/(mm)	22	Auxiliary magnetic pole arc coefficient	0.1
Air-gap length/(mm)	1	Turn-number per tooth of torque winding	50
Permanent magnet thickness/(mm)	2	Main magnetic pole remanence/(T)	1.23
Axial length/(mm)	40	Auxiliary magnetic pole remanence/(T)	1.23
Stator and rotor material	DW315-50	Turn-number per tooth of suspension winding	50

shows the main structural parameters of a BL-BLDCM-LTPMIMP.

Setting: The position when the main pole center is facing the positive direction of the horizontal X-axis is the rotor’s zero angle position. Then, for the magnetization method shown in Figure 2, choosing a pair of poles as the study object, the magnetization intensity can be decomposed into radial and tangential components, and the specific decomposition results are shown in

Table 2. Magnetization intensity of permanent magnet.

Magnetization Intensity Components	Angular Position to the Magnetic Pole Center
$\left\{\begin{array}{c}{M}_r={\displaystyle \frac{B_{r20}}{{\mu }_0}}\sin \left(\theta +{\displaystyle \frac{\pi }{2p}}\right)\\ M_{\theta }={\displaystyle \frac{B_{r20}}{{\mu }_0}}\cos \left(\theta +{\displaystyle \frac{\pi }{2p}}\right)\end{array}\right.$	$-{\displaystyle \frac{(2-\alpha )\pi }{2p}}<\theta <-\alpha {\displaystyle \frac{\pi }{2p}}$
$\left\{\begin{array}{c}{M}_r={\displaystyle \frac{B_{r10}}{{\mu }_0}}\\ M_{\theta }=0\end{array}\right.$	$-\alpha {\displaystyle \frac{\pi }{2p}}<\theta <\alpha {\displaystyle \frac{\pi }{2p}}$
$\left\{\begin{array}{c}{M}_r=-{\displaystyle \frac{B_{r20}}{{\mu }_0}}\sin \left(\theta -{\displaystyle \frac{\pi }{2p}}\right)\\ M_{\theta }=-{\displaystyle \frac{B_{r20}}{{\mu }_0}}\cos \left(\theta -{\displaystyle \frac{\pi }{2p}}\right)\end{array}\right.$	$\alpha {\displaystyle \frac{\pi }{2p}}<\theta <(2-\alpha ){\displaystyle \frac{\pi }{2p}}$
$\left\{\begin{array}{c}{M}_r=-{\displaystyle \frac{B_{r10}}{{\mu }_0}}\\ M_{\theta }=0\end{array}\right.$	$(2-\alpha ){\displaystyle \frac{\pi }{2p}}<\theta <(2+\alpha ){\displaystyle \frac{\pi }{2p}}$

. In Table 2, M_r and M_θ are the radial and tangential components of the magnetic field intensity, µ₀ is the vacuum permeability, p is the number of magnet pole pairs, α is the pole arc coefficient of the main pole, B_r10 is the remanent magnetization magnitude of the main pole, B_r20 is the remanent magnetization magnitude of the auxiliary pole, and the variable θ is the angular position of magnetic pole center, i.e., the rotor position angle.

To facilitate the analytical calculations, M_r and M_θ are expressed in the form of accumulated series as follows:

$$\left\{\begin{array}{l}{M}_r={\displaystyle \sum _{n=1,3,5\dots }^{\infty }{M}_{rn}\cos (np\theta )}\\ M_{\theta }={\displaystyle \sum _{n=1,3,5\dots }^{\infty }{M}_{\theta n}\sin (np\theta )}\end{array}\right.$$

(1)

In Equation (1), M_rn and M_θn are the nth harmonic component amplitudes of the radial and tangential components of magnetic field intensity, respectively.

Meanwhile, for a permanent magnet with pole pair number p, each pole occupies a mechanical angle of π/p along the circumference, and each pair of poles has a period of 2π/p. Then, M_rn and M_θn satisfy the following relationship within a pair of magnetic pole cycles:

$$\left\{\begin{array}{c}{M}_{rn}={\displaystyle \frac{2}{T}}{\displaystyle {\int }_0^T{M}_r\cos (np\theta )}d\theta \\ M_{\theta n}={\displaystyle \frac{2}{T}}{\displaystyle {\int }_0^T{M}_{\theta }\cos (np\theta )}d\theta \end{array}\right.$$

(2)

3.2. Analytical Calculation of the No-Load Magnetic Field

In this work, in order to simplify the analytical calculation of magnetic field distribution and make the modeling process clearer and more understandable, the analytical models of air-gap magnetic field are firstly established for the slotless BL-BLDCM-LTPMIMP, then by considering the slotting effect, the magnetic field solution model of the slotted BL-BLDCM-LTPMIMP is established, while at the same time, making assumptions as follows:

(1) Neglecting the magnetic saturation and assuming that the iron core has infinite permeability;

(2) Permanent magnets are linearly demagnetizing and are fully magnetized in the direction of magnetization.

Then, combining the Laplace equation and quasi-Poisson equation, and according to the continuous variation characteristics of boundary conditions, the magnetic field region of the whole motor can be divided into the air-gap magnetic field region I and the rotor permanent magnet magnetic field region II, as shown in Figure 3, where R_s is the inner diameter of the stator; R_r and R_m are the inner and outer diameters of the permanent magnet, respectively.

From the electromagnetism theory, within the air-gap region and the permanent magnet region, the magnetic induction intensity and the magnetic field intensity, respectively, satisfy the following relationship:

$${\mathit{B}}_{\mathit{r}\mathbf{{\rm I}}}={\mu }_0{H}_{\mathit{r}\mathbf{{\rm I}}},\hspace{1em} B_{\mathit{r}\mathbf{{\rm I}}\mathbf{{\rm I}}}={\mu }_0{\mu }_r{H}_{\mathit{r}\mathbf{{\rm I}}\mathbf{{\rm I}}}+{\mu }_{0}M$$

(3)

where B_rI and H_rI are the magnetic induction intensity and magnetic field intensity in the air-gap region, B_rII and H_rII are the magnetic induction intensity and magnetic field intensity in the permanent magnet region, and M is the magnetization intensity of the permanent magnet.

In the air-gap region I, the scalar magnetic potential of the magnetic field satisfies the Laplace equation; in the permanent magnet region II, the scalar magnetic potential of the magnetic field satisfies the Poisson equation. For this, there are the following relationship equations:

$${\displaystyle \frac{{\partial }^2{\phi }_{{\rm I}}}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial {\phi }_{{\rm I}}}{\partial r}}+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{\partial }^2{\phi }_{{\rm I}}}{\partial {\theta }^2}}=0$$

(4)

$${\displaystyle \frac{{\partial }^2{\phi }_{{\rm I}{\rm I}}}{\partial r^2}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial {\phi }_{{\rm I}{\rm I}}}{\partial r}}+{\displaystyle \frac{1}{r^2}}{\displaystyle \frac{{\partial }^2{\phi }_{{\rm I}{\rm I}}}{\partial {\theta }^2}}={\displaystyle \frac{1}{{\mu }_r}}divM$$

(5)

The radial and tangential components of the magnetic field intensity H can be expressed as follows:

$$H_r=-{\displaystyle \frac{\partial \phi }{\partial r}}, H_{\theta }=-{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \phi }{\partial \theta }}$$

(6)

In polar coordinates, the magnetization intensity M is written in the form of vector decomposition and summation as follows:

$$M=M_r{e}_r+M_{\theta }{e}_{\theta }$$

(7)

According to Equation (7), the divergence of the magnetization intensity M can be derived as follows:

$$divM={\displaystyle \frac{M_r}{r}}+{\displaystyle \frac{\partial M_r}{\partial r}}+{\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial M_{\theta }}{\partial \theta }}={\displaystyle \sum _{n=1,3,5\dots }^{\infty }\frac{1}{r}}{M}_n\cos (np\theta )$$

(8)

wherein,

$$M_n=M_{rn}+np{M}_{\theta n}$$

(9)

Under the condition of “np ≠ 1”, the general solution of Equations (4) and (5) can be obtained as follows:

$${\phi }_{{\rm I}}(r,\theta )={\displaystyle \sum _{n=1}^{\infty }(C_1{r}^{np}+C_2{r}^{-np})\cos (np\theta )}$$

(10)

$${\phi }_{{\rm I}{\rm I}}(r,\theta )={\displaystyle \sum _{n=1}^{\infty }(C_3{r}^{np}+C_4{r}^{-np})\cos (np\theta )}+{\displaystyle \sum \frac{M_{n}r}{{\mu }_r[1-(np)]}}\cos (np\theta )$$

(11)

where the coefficients C1, C2, C3, and C4 are determined by the boundary conditions of the magnetic field.

From the two-dimensional model of BL-BLDCM-LTPMIMP, the boundary conditions can be written as

$$\left\{\begin{array}{l}{H}_{\theta {\rm I}}(r,\theta ){|}_{r=R_s}=0\\ H_{\theta {\rm I}{\rm I}}(r,\theta ){|}_{r=R_r}=0\\ B_{r{\rm I}}(r,\theta ){|}_{r=R_m}=B_{r{\rm I}{\rm I}}(r,\theta ){|}_{r=R_m}\\ H_{\theta {\rm I}}(r,\theta ){|}_{r=R_m}=H_{\theta {\rm I}{\rm I}}(r,\theta ){|}_{r=R_m}\end{array}\right.$$

(12)

Combining the above differential equations and the magnetic field boundary conditions, the analytical calculation expressions for the radial component B_rI (r, θ) and tangential component B_θI (r, θ) of the air-gap magnetic flux density can be obtained as follows [15]:

$$\left\{\begin{array}{l}{B}_{r{\rm I}}(r,\theta )={\displaystyle \sum _{n=1,3,5\dots }^{\infty }{K}_B(n)f_{Br}(r)\cos (np\theta )}\\ B_{\theta {\rm I}}(r,\theta )={\displaystyle \sum _{n=1,3,5\dots }^{\infty }{K}_B(n)f_{B\theta }(r)\sin (np\theta )}\end{array}\right.$$

(13)

wherein:

$$f_{Br}(r)={\left({\displaystyle \frac{r}{R_s}}\right)}^{np-1}{\left({\displaystyle \frac{R_m}{R_s}}\right)}^{np+1}+{\left({\displaystyle \frac{R_m}{r}}\right)}^{np+1}$$

(14)

$$f_{B\theta }(r)=-{\left({\displaystyle \frac{r}{R_s}}\right)}^{np-1}{\left({\displaystyle \frac{R_m}{R_s}}\right)}^{np+1}+{\left({\displaystyle \frac{R_m}{r}}\right)}^{np+1}$$

(15)

$$K_B(n)={\displaystyle \frac{{\mu }_0{M}_n}{{\mu }_r}}{\displaystyle \frac{np}{[{(np)}^2-1]}}\cdot \left\{{\displaystyle \frac{(A_n-1)+2{(\frac{R_r}{R_m})}^{np+1}-(A_n+1){(\frac{R_r}{R_m})}^{2np}}{\frac{{\mu }_r+1}{{\mu }_r}[1-{(\frac{R_r}{R_s})}^{2np}]-\frac{{\mu }_r-1}{{\mu }_r}[{(\frac{R_m}{R_s})}^{2np}-{(\frac{R_r}{R_m})}^{2np}]}}\right\}$$

(16)

$$A_n={\displaystyle \frac{1}{np}}+{\displaystyle \frac{M_{rn}}{M_n}}(np-{\displaystyle \frac{1}{np}})$$

(17)

Substituting the parameters in Table 1 into the air-gap magnetic flux density expression in Equation (13), the analytical calculation expressions of the no-load air-gap magnetic field of the BL-BLDCM-LTPMIMP structure can be achieved.

Based on the analytical computational model derived above, the radial air-gap magnetic density of the slotless BL-BLDCM-LTPMIMP can be solved. Firstly, substituting the radial and tangential components of magnetic field intensity at different positions within the range of a pair of magnetic poles in Table 2 into Equation (2), and M_rn and M_θn can be obtained by integral operation; then, M_n can be obtained by substituting M_rn and M_θn into Equation (9), which in turn leads to A_n in Equation (17); Next, f_Br and K_B(n) can be obtained by substituting the basic structural parameters of BL-BLDCM-LTPMIMP, as well as the calculation results obtained above into Equations (14) and (16); finally, the radial air-gap magnetic density of a slotless BL-BLDCM-LTPMIMP can be derived from Equation (13).

In this work, the ontology model of a BL-BLDCM-LTPMIMP structure is established by finite element simulation software Ansys-Maxwell 2022 R1. Firstly, the basic structure model of a brushless DC motor is designed with the help of the RMxprt module; then, according to the characteristics of the proposed BL-BLDCM-LTPMIMP structure, the wrapping pattern of two sets stator windings (i.e., torque windings and suspension windings), and the structures of the permanent magnets are modified; and finally, the finite element analysis models of the slotless BL-BLDCM-LTPMIMP and slotted BL-BLDCM-LTPMIMP are obtained, respectively. The stator and rotor materials are shown in Table 1. Before the FEM analysis being performed, the inside selection, on selection, and surface approximation settings are required to complete the mesh sectioning of BL-BLDCM-LTPMIMP. In this work, a 1.0 mm long air-gap region is dissected into eight layers. The mesh sectioning of the slotless and slotted motors are shown in Figure 4a,b. The boundary condition is set to “natural boundary condition”.

The above solved analytical calculation expressions of the no-load air-gap magnetic field are obtained under the assumption conditions that the inner surface of the stator is smooth. That is to say, the effect of the tooth slot on the air-gap magnetic field is neglected for the time being. Therefore, when verifying the accuracy of the analytical calculation expressions of this air-gap magnetic flux density by FEM, the finite element analysis model of the BL-BLDCM-LTPMIMP is firstly modified to a slotless motor (the windings are placed on the inner surface of the stator core). In the case of the same motor structure parameters, at the moment when the permanent magnet N-pole is facing the positive X-axis direction, the comparison waveforms of the radial air-gap magnetic flux density obtained by the analytical calculation method and the FEM are shown in Figure 5. As shown in Figure 5, the following research results are obtained:

(1) For the slotless BL-BLDCM-LTPMIMP, the no-load air-gap magnetic flux density waveforms obtained by the analytical calculation method basically overlap completely with that by the FEM. Then, the analytical calculation equations of the no-load air-gap magnetic flux density of the slotless BL-BLDCM-LTPMIMP derived in this work are valid and accurate.

(2) When adopting the proposed BL-BLDCM-LTPMIMP structure, the expected shoulder-shrugged trapezoidal wave air-gap magnetic field can be obtained.

The previous analysis is based on the assumption that the stator is not slotted. But in practice, the armature windings are generally placed in the stator slots, and at this point, it is not possible to ignore the impact of the slotting effect on the distribution of the air-gap magnetic field, which also causes the motor to produce cogging torque.

In order to obtain the analytical calculation equations of the no-load air-gap magnetic flux density of the slotted BL-BLDCM-LTPMIMP, this work introduces the method of relative permeability function to solve the problem. The relative permeability function of a slotted motor can be expressed as follows [15]:

$$\lambda (\theta ,r)=\left\{\begin{array}{c}{\mathrm{\Lambda }}_0\left[1-\beta (r)-\beta (r)\cos {\displaystyle \frac{\pi }{0.8{\alpha }_0}}\theta \right], 0\le \theta \le 0.8{\alpha }_0\\ {\mathrm{\Lambda }}_0, 0.8{\alpha }_0\le \theta \le {\displaystyle \frac{{\alpha }_t}{2}}\end{array}\right.$$

(18)

$$\beta (r)={\displaystyle \frac{1}{2}}\left[1-1/\sqrt{1+{({\displaystyle \frac{b_0}{2g^{\prime }}})}^2(1+v^2)}\right]$$

(19)

In Equations (18) and (19), b₀ is the slot width; α₀ is the angle occupied by the slot width over the entire circle circumference, α₀ = b₀/R_s; Q_s is the number of stator slots; α_t is the groove pitch angle, α_t = 2π/Q_s; g is the length of the air-gap; g′ is the equivalent air-gap length, g′ = g + h_m/µ_r; h_m is the thickness of the permanent magnet; ν is the calculation process variable. For an inner rotor structure AC motor, ν can be obtained by solving the following equation [15]:

$$y{\displaystyle \frac{\pi }{b_0}}={\displaystyle \frac{1}{2}}\ln \left[{\displaystyle \frac{\sqrt{a^2+v^2}+v}{\sqrt{a^2+v^2}-v}}\right]+{\displaystyle \frac{2g^{\prime }}{b_0}}\arctan ({\displaystyle \frac{2g^{\prime }}{b_0}}{\displaystyle \frac{v}{\sqrt{a^2+v^2}}})$$

(20)

$$a^2=1+{(2g^{\prime }/b_0)}^2$$

(21)

where y is the distance from the outer diameter of the rotor core to the air-gap r, y = r − (R_s − g′), and R_s is the inner diameter of the stator core.

By joining the above equations, the relative permeability function can be obtained as follows:

$$\overline{\lambda }(\theta ,r)={\displaystyle \frac{\lambda (\theta ,r)}{{\mathrm{\Lambda }}_0}}={\displaystyle \frac{\lambda (\theta ,r)}{{\mu }_0/g^{\prime }}}$$

(22)

For ease of solution, the above equation is written in the form of a Fourier series:

$$\overline{\lambda }(\theta ,r)={\displaystyle \frac{{\mathrm{\Lambda }}_0}{2}}+{\displaystyle \sum _{n=1}^{\infty }{\mathrm{\Lambda }}_n(r)\cos (n{Q}_s\theta )}$$

(23)

wherein,

$${\mathrm{\Lambda }}_0={\displaystyle \frac{2}{{\alpha }_t}}{\displaystyle \underset{-{\alpha }_t/2}{\overset{{\alpha }_t/2}{\int }}\overline{\lambda }(\theta ,r)d\theta }={\displaystyle \frac{4}{{\alpha }_t}}\left\{{\displaystyle \underset{0}{\overset{0.8{\alpha }_0}{\int }}\left(1-\beta (r)-\beta (r)\cos \frac{\pi }{0.8{\alpha }_0}\theta \right)d\theta +{\displaystyle \underset{0.8{\alpha }_0}{\overset{0.5{\alpha }_t}{\int }}d\theta }}\right\}$$

(24)

$$\begin{array}{ll}{\mathrm{\Lambda }}_n(r)& ={\displaystyle \frac{2}{{\alpha }_t}}{\displaystyle \underset{-{\alpha }_t/2}{\overset{{\alpha }_t/2}{\int }}\overline{\lambda }(\theta ,r)\cos (n{Q}_s\theta )d\theta }\\ & ={\displaystyle \frac{4}{{\alpha }_t}}\left\{{\displaystyle \underset{0}{\overset{0.8{\alpha }_0}{\int }}\left(1-\beta (r)-\beta (r)\cos \frac{\pi }{0.8{\alpha }_0}\theta )\cos (n{Q}_s\theta )\right)d\theta }+{\displaystyle \underset{0.8{\alpha }_0}{\overset{0.5{\alpha }_t}{\int }}\cos (n{Q}_s\theta )d\theta }\right\}\end{array}$$

(25)

Therefore, for the slotted BL-BLDCM-LTPMIMP in this work, the radial and tangential air-gap magnetic densities can be expressed as follows:

$$\left\{\begin{array}{l}{B^{\prime }}_{r{\rm I}}(r,\theta )=B_{r{\rm I}}(r,\theta )\cdot \overline{\lambda }(\theta ,r)\\ {B^{\prime }}_{\theta {\rm I}}(r,\theta )=B_{\theta {\rm I}}(r,\theta )\cdot \overline{\lambda }(\theta ,r)\end{array}\right.$$

(26)

After solving the relative permeability function, the radial air-gap magnetic density of the slotted BL-BLDCM-LTPMIMP can be solved from the above analytical calculation model, the specific process is as follows:

(1) According to Equation (13), solve the radial air-gap magnetic density of the slotless BL-BLDCM-LTPMIMP (methods or steps are the same as before);

(2) Solve the relative permeability function. Firstly, the process variable ν is solved from Equations (20) and (21); after substituting ν into Equation (19), β(r) is obtained; and according to Equations (24) and (25), Λ₀ and Λ_n are obtained, respectively; then, based on the Fourier expansion series expression in Equation (23), the relative permeability function can be obtained.

(3) According to Equation (26), the radial air-gap magnetic density data of the slotted BL-BLDCM-LTPMIMP can be obtained by the corresponding multiplication equation between the calculation results of Equation (23) and those of Equation (13).

The radial air-gap magnetic density waveforms considering the slotting effect are as shown in Figure 6, while Figure 7 shows the permanent magnetic density distribution of slotted BL-BLDCM-LTPMIMP. As shown in Figure 6 and Figure 7, the following research results are obtained:

(1) The waveforms of the air-gap magnetic flux density obtained by the analytical calculation method are basically consistent with that by the FEM, and the air-gap magnetic field still has the desired shoulder-shrugged characteristics. But there is a slight difference in the part of the tooth groove edges, i.e., the air-gap magnetic flux density solved by the FEM is slightly increased at the tooth edge. The difference is due to the magnetic field aggregation effect at the edge of the tooth groove, which causes part of the magnetic field to aggregate at the tooth tip, and the analytical calculation model has still not taken the magnetic field aggregation effect into account for the time being.

(2) In other positional regions, the air-gap magnetic flux density waveforms obtained by the two methods basically coincide.

(3) Overall, the waveform error of the air-gap magnetic flux density obtained by the two methods is very small and within the allowable range. And then, under the conditions of considering the slotting effect, the derived analytical calculation model of no-load air-gap magnetic flux density of the BL-BLDCM-LTPMIMP is valid and has high accuracy.

3.3. Analytical Calculation of Synthetic Air-Gap Magnetic Fields

During the operation of BL-BLDCM-LTPMIMP, three magnetic fields exist in the air-gap, including the permanent magnet field, the armature reaction field of the torque winding, and that of the suspension winding.

We set the mechanical angular velocity of the rotor as ω, assuming that the position of the permanent magnet shown in Figure 1 is the initial position at time zero. Then, the distribution of the magnetic field generated by the permanent magnet in space can be expressed as

$$B_1(\theta ,t)={\displaystyle \sum _{n=1,3,5\dots }^{\infty }{K}_B(n)f_{Br}(r)\cos \left[np\left(\theta -\omega t-\frac{\pi }{3}\right)\right]}$$

(27)

Here, the independent suspension windings of BL-BLDCM-LTPMIMP are used to control the radial suspension force, and the conduction phase of the suspension windings is determined according to the location region of the rotor. The conduction conditions of three-phase suspension windings at different position regions of the rotor are shown in

Table 3. Conductive phase of suspension winding with rotor position.

Position Angle (°)	Conductive Winding
0–15	U1− U2−
15–45	W1− W2−
45–75	V1− V2−
75–105	U1+ U2+
105–135	W1+ W2+
135–165	V1+ V2+
165–180	U1− U2−

.

According to the working principle of BL-BLDCM-LTPMIMP, the suspension current injected into the suspension winding is a square wave that varies with the rotor position. Under the action of the square-wave suspension magnetomotive force, a square-wave suspension magnetic field B₂(θ, t) with step jump rotation will be generated in the air-gap. In fact, the actual suspension magnetic field has transition regions that are not strictly square-wave, and approximation is carried out here. When a current with magnitude I is injected into the suspension winding, the amplitude of the generated square-wave magnetic field can be expressed as

$$B_m={\mu }_{0}F/g^{\prime }$$

(28)

where F = NI is the magnetomotive force generated by the single tooth suspension current energized, and g′ is the equivalent air-gap length.

Taking the moment of the zero initial position of the rotor shown in Figure 1 as an example, it is assumed that the current flowing through the U1 phase winding is I₁ and the current flowing through the U2 phase winding is I₂. Then, the special distribution of the suspension magnetic field is shown in

Table 4. Suspension magnetic field distribution.

Position Angle (°)	Size of Magnetic Field
0–45	0
45–75	${\mu }_{0}N{I}_1/g^{\prime }$
75–135	0
135–165	${\mu }_{0}N{I}_2/g^{\prime }$
165–225	0
225–255	$-{\mu }_{0}N{I}_1/g^{\prime }$
255–315	0
315–345	$-{\mu }_{0}N{I}_2/g^{\prime }$
345–360	0

. After the suspension magnetic field is superimposed with the permanent magnetic field, the synthetic air-gap magnetic field under the action of the permanent magnet and the suspension winding in the air-gap can be obtained. Figure 8 shows the spatial distribution of the synthetic magnetic density produced together by the permanent magnet and the suspension winding. Figure 9 shows the comparison waveforms of the synthetic air-gap magnetic density obtained from the FEM and analytical calculation method.

The torque winding of the BL-BLDCM-LTPMIMP is in the “three-phase, two-conduction, six-state” control mode. The conduction conditions of three-phase torque winding at different positional regions of the permanent magnet rotor are shown in

Table 5. Conductive phase of torque winding with rotor position.

Position Angle (°)	Conductive Winding
0–15	C+ B−
15–45	A+ B−
45–75	A+ C−
75–105	B+ C−
105–135	B+ A−
135–165	C+ A−
165–180	C+ B−

.

Similarly, when the rotor is at zero initial position, B- and C-phase torque windings then conduct. Assuming that the amplitude of the injected torque current into the torque windings is I₃, the torque magnetic field B₃(θ, t) generated in space is still a jump-rotation (approximate) square-wave magnetic field, and the angle of the circumferential space spanned by the flat-top of the waveform is 60 degrees. The special distribution of the torque winding magnetic field is shown in

Table 6. Torque magnetic field distribution.

Position Angle (°)	Magnetic Field Size
0–45	$-{\mu }_{0}N{I}_3/g^{\prime }$
45–75	0
75–135	${\mu }_{0}N{I}_3/g^{\prime }$
135–165	0
165–225	$-{\mu }_{0}N{I}_3/g^{\prime }$
225–255	0
255–315	${\mu }_{0}N{I}_3/g^{\prime }$
315–345	0
345–360	$-{\mu }_{0}N{I}_3/g^{\prime }$

.

When the BL-BLDCM-LTPMIMP is energized with the 5A torque current and the 2.5A suspension current, the synthetic magnetic field of the permanent magnet, suspension winding and torque winding is shown in Figure 10. While Figure 11 shows the spatial distribution of the magnetic density produced by the permanent magnet, torque winding and suspension winding together. As shown in Figure 8, Figure 9, Figure 10 and Figure 11, we obtained the following findings:

(1) The synthetic air-gap magnetic field waveforms obtained by the FEM and the analytical calculation method are basically matches. In the steady-state region, the error between them remains within a small range of 1.5%, indicating that the analytical calculation models of the air-gap-synthesized magnetic field is accurate and reliable. The reason for the small error lies in that some approximations were made in the analytical modeling of armature reaction magnetic fields for torque winding and suspension winding.

(2) There is a peak calculation difference of approximately 5.1% in the transition region near the edge of the tooth groove. The reason for the peak difference is that the magnetic field aggregation effect is not considered when modeling using the analytical method.

(3) Under the effect of suspension magnetic field, the magnetic field appears to have an asymmetric spatial distribution, the magnetic field in some spatial regions is strengthened, and the magnetic field in the spatially symmetric regions is suppressed, which is in accordance with the working principle of the bearingless motor [9,26].

4. Comparison of Characteristics with Ordinary BL-BLDC Motor

In order to verify the performance characteristics of the proposed BL-BLDCM-LTPMIMP, FEM computations under the same conditions are made to compare with the ordinary BL-BLDCM with single-radial magnetization. The permanent magnet pole-arc coefficient of ordinary BL-BLDCM is selected as 0.85, and the main magnetic pole of BL-BLDCM-LTPMIMP is selected to be the same as ordinary BL-BLDCM, i.e., α = 0.85; other structural parameters are shown in Table 1.

We set the simulation conditions, e.g., rated speed is 1500 r/min (i.e., 20 ms corresponds to 180-degree mechanical angle); take the generation of positive torque and suspension force along the x-axis as an example; and apply the same currents to both motors, including 5 A torque current and 2.5 A suspension current (here, the suspension current of U1 is used as a reference). The FEM simulation comparation waveforms of no-load radial air-gap flux density and cogging torque are given in Figure 12 and Figure 13, respectively. Figure 14 shows the suspension current waveform. The FEM simulation comparison waveforms of electromagnetic torque and radial magnetic suspension force are given in Figure 15 and Figure 16, respectively. For BL-BLDCM-LTPMPIMP, Figure 17 gives the waveforms of torque currents and back electromotive force (EMF) of the three-phase torque winding. According to Figure 12, Figure 13, Figure 14, Figure 15, Figure 16 and Figure 17, the specific research results are as follows:

(1) The air-gap magnetic fields of BL-BLDCM-LTPMIMP and ordinary BL-BLDCM are both affected by the slotting effect.

(2) Compared to the cogging torque amplitude 58 mN m of ordinary BL-BLDCM, the cogging torque amplitude of the BL-BLDCM-LTPMIMP is about 53 mN·m and has a reduction of about 8.62%. The smaller cogging torque basically does not have any noticeable effect on the electromagnetic torque waveform.

(3) Compared with the ordinary BL-BLDCM structure, the proposed BL-BLDCM-LTPMIMP structure can significantly enhance the magnetic flux densities at both ends of the flat-top region of the air-gap magnetic field and obtain a shoulder-shrugged trapezoidal wave air-gap magnetic field, which is advantageous for reducing the commutation torque pulsation. The proposed BL-BLDCM-LTPMIMP has improved the average torque by about 3.13% and reduced the peak-to-peak torque pulsation ratio by about 2.06% (from 11.15% to 9.09%).

(4) Compared to the ordinary BL-BLDCM, the average radial suspension force of BL-BLDCM-LTPMIMP is basically unchanged, and the peak-to-peak pulsation ratio increased by 0.70% (from 5.50% to 6.20%). Smaller suspension force pulsation ratios do not affect the rotor suspension control obviously.

(5) In the case of BL-BLDCM-LTPMIMP using a single-tooth-concentrated (short distance) winding, the trapezoidal feature of its EMF waveform is not obvious (or the flat-top area of the trapezoid is relatively narrow), which is different from the common brushless DC with full-pitch windings. Due to the influence of winding electromagnetic inertia and commutation delay, the torque current is delayed by about 7 degrees (0.4 ms) compared to EMF. However, during the stable operation of BL-BLDCM-LTPMIMP, the sum of the active power of the three-phase windings remains basically unchanged. For example, for the three different moments of 1.0 ms, 4.0 ms, and 8.0 ms selected optionally in Figure 17, the active power sum of three-phase windings can be calculated as 235.45 W, 232.44 W, and 235.58 W, respectively, and the deviation between them is very small. The motor’s active power remains essentially constant, which is beneficial for reducing torque ripple.

5. Conclusions

In this work, to solve the problem of large commutation torque ripple of ordinary BL-BLDCM, a novel BL-BLDCM-LTPMIMP structure is proposed, and it was modeled using the analytical calculation method. Firstly, the topology and working principle of the BL-BLDCM-LTPMIMP are introduced. Then, the analytical formulas of air-gap magnetic field for the BL-BLDCM-LTPMIMPs are derived, and the armature reactive magnetic fields of the torque winding and suspension winding are also calculated analytically. Finally, by means of FEM, the electromagnetic characteristics of the proposed BL-BLDCM-LTPMIMP structure are compared with those of ordinary BL-BLDCM, and the analytical calculation models of the BL-BLDCM-LTPMIMP are proved. The research results can be concluded as follows:

(1) Compared with the ordinary BL-BLDCM structure, the BL-BLDCM-LTPMIMP structure proposed in this work can achieve a shoulder-shrugged trapezoidal wave air-gap magnetic field, and it can obtain a larger steady-state torque and a higher torque dynamic stability (i.e., lower torque pulsation ratio). For the two BL-BLDCM structures, the obtained steady state values of the radial magnetic levitation force are basically equal, and the pulsation ratios are all within acceptable range.

(2) The analytic calculation models of the proposed BL-BLDCM-LTPMIMP are valid and correct. In the slotless case, the calculation results obtained from the FEM and the analytical calculation method are essentially identical, while in the slotting case, in view of the magnetic field aggregation effect at the edge of the tooth groove, the armature reaction magnetic field pulsations caused by winding current commutation and other factors, the calculation results obtained by analytical methods still have some errors, but the errors are very small.

The research results in this work can provide certain theoretical basis or model support for the air-gap magnetic field calculation and structure optimization of the BL-BLDCM. Of course, the saturation of the magnetic circuit will have an effect on the air-gap magnetic field distribution of a BL-BLDCM-LTPMIMP to some extent. Due to the length limitation of the paper, for the BL-BLDCM-LTPMIMP in this work, the analytical calculation method of air gap magnetic field, which takes into account the magnetic field aggregation effect of stator tooth tip and the effect of magnetic circuit saturation, as well as the analytical calculation method of the electromagnetic torque and radial magnetic suspension force, will be given gradually in the subsequent studies. In addition, for the BL-BLDCM-LTPMIMP in this work, the multi-objective joint optimization of structural parameters with the goal of improving the electromagnetic torque and radial magnetic suspension force control characteristics, and the energization strategy of stator windings with the goal of energy saving and consumption reduction, are also issues to be further studied.