Analytical Model of Modular Spoke-Type Permanent Magnet Machines for In-Wheel Traction Applications

This article proposes an analytical model of modular spoke-type permanent magnet (MSTPM) machines based on air-gap field modulation (AFM) theory. Firstly, a fundamental AFM model of open-circuit MSTPM machines is introduced. The open-circuit air-gap field of MSTPM machines is determined by three fundamental elements including the primitive magnetizing magnetomotive force (MMF) produced by permanent magnet (PM), and two modulators which consist of stator and rotor permeance. The analytical MMF excited by PM (PM-MMF) can be calculated by using magnetic circuit method, while the stator and rotor permeance models are developed based on relative permeance (RP) method. Thereafter, a general model is proposed to calculate the open-circuit back electromotive force (EMF) of MSTPM machines. Further, the winding inductance model is established on the basis of equivalent magnetic circuit method and RP model. Finally, the machine performance is predicted by the analytical model, and verified by both finite element analysis (FEA) and experimental results.


Λr1
Amplitude of fundamental relative rotor permeance

Λs(θ, t)
Relative stator permeance at (position θ, moment t) Br(θ, t) Radial air-gap flux density at (position θ, moment t) B ho The h o th harmonic of air-gap flux density ec (t) Back-EMF of a single coil at moment t e k The k th phase back-EMF N recent years, in-wheel traction (IWT) machines have been increasingly hot research topics due to the high torque density, and excellent system efficiency [1]- [3]. It is believed that the electric vehicles (EVs) driven by IWT machines are ideal candidates of future transportation. According to the available literature, a variety of machine topologies have been designed for the IWT applications, including the switched reluctance (SR) machines [4], the surface permanent magnet (SPM) machines [5], the V-shaped interior permanent magnet (IPM) machines [2], the flux-switching permanent magnet (FSPM) machines [1], the magnetic gear (MG) machines [6], etc.
In [7], [8], the MSTPM machines are found to feature a higher machine efficiency, a better flux-weakening performance, and a stronger over-load capability than conventional flux-switching permanent magnet (FSPM) I machines, and surface-mounted permanent magnet (SPM) machines used in IWT applications. It is certain that the MSTPM machines deserve more research attentions, especially in the field of IWT applications.
By now, an analytical prediction of air-gap field in MSTPM machines has been derived based on conformal mapping method [7]. The design considerations of the MSTPM machines were discussed in [8], where the elimination of even-order harmonics and unbalanced magnetic force, and the optimization of rotor tooth width were involved. Further, the back-EMF waveform, and possible stator-slot/rotor-pole combinations of MSTPM machines were analyzed in [9]. The influence of magnetization mode on machine performance was investigated in [10], [11]. However, a general analytical model of MSTPM machines has not been built in existing literature, with magnetic field, back-EMF, and winding inductance included.
On the other hand, the AFM theory has been used in the analysis of a variety of kinds of electrical machines [12], [13], including FSPM machines [14], [15], flux-reversal permanent magnet (FRPM) machines [16], [17], SPM machines [18], fractional-slot concentrated-winding (FSCW) vernier permanent magnet (VPM) machines [19], partitioned-stator FSPM machines [20], etc. A general AFM theory has been built in [12], where a basic unit motor is normalized into a cascade of three key elements, namely the excitation source (primitive magneto-motive force), the modulator (salient pole/short circuit coil/flux guide) and the filter (armature winding). It has been proved that, the general AFM theory is valid for all electromagnetic machines, irrespective of DC or AC machines, salient or non-salient machines, sinusoidal or rectangular current driven machines, etc., with respect to both the qualitative and quantitative analyses associated with clear physical meanings. Further, the AFM theory has also been successfully used in the topology exploration of novel electrical machines [20], [21]. Hence, it is certain that an analytical model of the MSTPM machines can be built based on the general AFM theory.
Since the MSTPM machines have different structures from traditional electrical machines, and different magnetization modes of MSTPM machines lead to different stator/rotor structures, resulting in different winding inductance formulae [23], [24], it is necessary to build the winding inductance model to guide the future design of MSTPM machines. Besides, the MSTPM machine is one type of FSCW machines, of which the main winding inductance is the slot leakage component and air-gap component [25], [26]. Hence, special attention should be focused on the slot leakage and air-gap inductances of MSTPM machines. This paper is arranged as follows. A mathematical model of air-gap open-circuit field in MSTPM machines with different magnetization modes is set up on the basis of AFM theory and RP method in Section II. Then, an analytical model of phase back-EMF is developed in Section III, where the relationships between spatial field harmonics and temporal back-EMF harmonics are investigated. Thereafter, the winding inductance of the proposed machines is analyzed in Section III. Further, the machine performance is analytically predicted according to the foregoing models. Finally, two-dimensional (2D) finite element analysis (FEA) projects are built, and corresponding prototyped machines are assembled and tested to verify the analytical results.

II. AFM IN MSTPM MACHINES
In this section, the AFM model of MSTPM machines is built. Some assumptions should be obeyed: 1) no 3D end-effect; 2) the infinite permeability of silicon steel sheets; 3) the same permeability of both magnets and nonmagnetic support cells as vacuum; 4) ignoring tangential air-gap field component.

A. Machine Topology
As shown in Fig. 1, there are two different magnetization modes for MSTPM machines, which in this paper are referred to as M-I and M-II modes. The only difference is that the magnetization directions of two adjacent magnets are same in M-I machines, and reversal in M-II machines. Correspondingly, Figs. 2(a) and 2(b) show the PM flux paths of these two magnetization modes.
It can be found that under M-I mode, the magnetic polarity reverses in every rotor tooth, namely N-S-N-S…, whereas under M-II mode, the magnetic polarity reverses in every two rotor teeth namely S-N-N-S…, which results in distinct models as follows. Hence, the M-I MSTPM machines are equivalent to the topology shown in Fig

B. General AFM Model
From Figs. 2(c) and 2(d), it can be observed that the open-circuit air-gap field of MSTPM machines is due to the modulation of the primitive PM-MMF by both rotor and stator. Hence, the PM-MMF, the stator permeance, and the rotor permeance are investigated in this paper, respectively.
The MSTPM machines is one kind of rotor-magnet (RM) FSCW machine where magnets are housed on the rotor, which means the PM-MMF is rotary. Based on Fourier series, the PM-MMF is expressed by where p e is pole-pair number of PM-MMF,  is mechanical rotational speed,  is air-gap position, and t is time.
For M-I machines, the pole-pair number pe is 2 For M-II machines, p e yields 2 3 Further, the relative stator permeance of the MSTPM machines can be expressed as where N st is the number of stator slots.
In [9], it is found that the pole-pair number pe and the rotor tooth number N rt of M-I MSTPM machines should be constrained by (5) in order to achieve maximum torque.

5
As for M-II MSTPM machines, according to [11], it is found that the non-magnetic parts, which are sandwiched by rotor teeth, should be removed to achieve the maximum torque. Hence, in this paper, the analysis of the M-II MSTPM machines is based on the optimal structure shown in Fig. 3. Therefore, the pole-pair number pe and the rotor tooth number Nrt of the M-II MSTPM machines is constrained by (5) as well. C. PM-MMF In the AFM model of MSTPM machines, the PM-MMF is the square wave shown in Fig. 4, which is different from the conventional MMF-permeance method [8], [11]. In the conventional MMF-permeance method, the PM-MMF waveform always carries the harmonic information of rotor structure by assigning the width of wave crest of PM-MMF. However, in the AFM model, the harmonic information of rotor structure is carried by the relative rotor permeance. The PM-MMF waveform only carries the information on amplitude and pole-pair of the MSTPM machines. The amplitude of the PM-MMF FPM can be calculated by using equivalent magnetic circuit model of two types of MSTPM machines in [11]. Compared to the conventional MMF-permeance method, the proposed model provides more accurate information on field harmonics because of the more accurate rotor information. In Fig. 5, the air-gap field harmonic spectra of an M-I MSTPM machine are calculated by the FEA model, the MMF-permeance method, and the proposed model, respectively, while the corresponding calculation errors of the MMF-permeance method and the proposed model are shown in Fig. 6. It is found that the results calculated by the MMF-permeance method feature a bigger error.

D. Relative Stator Permeance
Since only the radial flux-density component is considered in the AFM model, a RP model rather than a complex RP model is built, decreasing the complexity of stator permeance model [27]- [31]. As shown in Fig. 7, the stator slot is assumed as an infinite deep rectilinear slot. According to [32], at the center circle of air-gap, the RP of the infinite deep stator slot is Based on equations (8)-(13), the relative stator permeance in the half stator pitch can be calculated. Then the whole stator permeance can be extended based on the symmetric condition.
The single stator pitch of the MSTPM machines with Q stator-slot is 2 14

E. Rotor Permeance
Similarly, the rotor slot is also assumed as an infinite deep rectilinear slot as shown in Fig. 8 Based on equations (15)- (20), the relative rotor permeance in the half rotor pitch can be calculated, and the whole rotor permeance is extended according to the geometric symmetry.
For the MSTPM machines with p e pole-pair PM-MMF, the single rotor pitch is where Nc is the number of turns per coil, r is the radius of the center circle of air-gap, and l is the effective stack length. Based on (7), the coil back-EMF of MSTPM machines is From (23), it is found that the back-EMF frequency in MSTPM machines is determined by the pole-pair number pe, and the rotor mechanical speed . Further, each harmonic of coil back-EMF is induced by different spatial field harmonics, which means the back-EMF in MSTPM machines is caused by AFM phenomenon. The relationship between the spatial field and temporal back-EMF harmonics in MSTPM machines is determined by the pole-pair number p e , stator tooth number N st , and rotor tooth number Nrt.
Consequently, the electrical speed ω e of coil back-EMF is

24
Hence, even-order harmonics will not appear in the back-EMF of ideal MSTPM machines whose PM-MMF waveforms are square waves. However, some even-order harmonics still exist in MSTPM machines caused by the permeability difference between magnets and nonmagnetic support cells, as reported in [8]. Since the magnets and nonmagnetic support cells have been assumed with the same permeability as vacuum, the even-order harmonics are ignored in this paper.
The relationship between spatial and temporal harmonics is concluded in Table I. The rotation direction of back-EMF harmonic is determined by multiplying electrical speed and field harmonic order. where kp is the winding pitch factor, kd is the winding distribution factor, B ho is the amplitude of the ho th flux density harmonic. The total k th back-EMF harmonic e k is 26

IV. WINDING INDUCTANCE IN MSTPM MACHINES
Since the MSTPM machines belongs to the FSCW-PM machines, the winding inductance can be divided into two parts: 1) slot leakage component; 2) air-gap component.

F. Slot Leakage Component
As the MSTPM machines for IWT applications are equipped with external rotor, the stator slot is normalized as the tapered slot shown in Fig. 9. In FSCW machines with double-layer windings, each winding slot is made up of two conductors shown in Fig. 9(a), namely as conductor 1 and conductor 2. In FSCW machines with single-layer windings, each winding slot is made up of one conductor shown in Fig. 9(b), namely as conductor 3. For both double-and single-layer FSCW machines, calculating the leakage permeance of single slot is the key step in the prediction of slot leakage inductance. Each armature slot can be divided into three regions, i.e., the conductor region, the trapezoid region, and the rectangular region, whose height is d1, d2, and d3, respectively.
For the conductor region whose height is d 1 , the leakage permeance is [33] For the trapezoid region whose height is d 2 , the leakage permeance is ln 28 For the rectangular region whose height is d 3 , the leakage permeance is

29
For single-layer machines, the slot leakage self-inductance per phase is 2 30 The slot leakage mutual-inductance per phase is

31
For double-layer machines with odd phase number m, the slot leakage self-inductance per phase is 4 4 32 The slot leakage mutual-inductance per phase is

33
For double-layer machines with even phase number m, the slot leakage self-inductance per phase is 4 2 34 The slot leakage mutual-inductance per phase is

35
where Nsub is the number of the sub-machine that a machine can be divided into, being the greatest common divisor (GCD) of Q and pe, as expressed by , 36

G. Air-gap Component
The air-gap component consists of two parts which are average inductance and position-varying inductance. The average component can be obtained on the basis of the average relative rotor and stator permeance.
The average component of air-gap self-inductance is

38
For the position-varying component of self-inductance, only the fundamental position-varying inductance is considered, whose spatial order is ℎ 39 Then, the amplitude of position-varying component can be calculated by

Λ Λ 40
where Λ r1 is the fundamental amplitude of the relative rotor permeance. Hence, the self-inductance per phase is sin 41 Since the mutual-inductance is much less than the self-inductance, only the average mutual-inductance is considered in this paper. In FSCW machines with single-layer windings, the air-gap mutual-inductance is 0 42 In FSCW machines with double-layer windings, the average air-gap mutual-inductance is Finally, the phase-to-phase mutual inductance is In this section, the electromagnetic performance of two M-I and M-II MSTPM machines for IWT applications are predicted by the proposed analytical model. Since the saturation phenomenon is ignored, the on-load performance is predicted according to the analytical open-circuit results. Then, the analysis is verified by both the FEA-predicted and experimental results. The key parameters are listed in Table II.

A. Experimental Platform
In Figs. 10 and 11, the 48s/52p M-I and 24s/26p M-II MSTPM machines are illustrated. In both machines, rotor modules are inserted into wheel rims using dovetail slot structure, which results in a high integrity. The interstices between the modular rotors and the rim are filled by anaerobic adhesive to improve structural stability.
As shown in Fig. 12, the experimental platform is made up of the prototype machine, the load (a magnetic powder brake), the digital oscilloscope (Agilent MSO-X 3024A), the three-phase permanent-magnet synchronous machine (PMSM) controller, and the DC power supply (KIKUSUI PAT250-32T).

B. Open-Circuit Air-gap Flux Density
As described in the aforementioned analytical model, the 48-slot/52-pole (48s/52p) M-I MSTPM machine is normalized as the modulation of the PM-MMF, the rotor permeance, and the stator permeance shown in Figs. 13-15. The amplitude of the PM-MMF is derived based on the equivalent magnetic path model, which is 490.6 A. The air-gap flux density is further calculated by (7), as shown in Fig. 16. The analytical results are compared with the 2D-FEA-predicted results in Fig. 17 and Table III. According to Table I, the 26-th and 74-th spatial harmonics are the dominant harmonics inducing the fundamental back-EMF in 48s/52p MSTPM machines. It is found that the analytical 26-th harmonic amplitude is very close to the 2D-FEA results, where the error is less than 1%. Although analytical and FEA-predicted 74-th harmonics are different, this difference is acceptable since they only produce little amount of fundamental back-EMF.         Fig. 21. The analytical results are compared with the 2D-FEA-predicted results in Fig. 22 and Table IV. According to Table I, the 13-th and 37-th spatial harmonics are the dominant harmonics inducing the fundamental back-EMF in 24s/26p MSTPM machines. It is found that the analytical 13-th harmonic amplitude is close to the 2D-FEA results as well. Also, the difference between analytical and FEA-predicted back-EMF harmonics is acceptable.

C. Phase Back-EMF and Inductance
According to formulae (25) and (26), and Table I, the phase back-EMF can be analytically calculated. In this paper, back-EMF harmonics are ignored in analytical prediction since proposed machines are AC machines, where only fundamental back-EMF can produce electromagnetic torque.
The analytical, FEA-predicted, and tested phase to phase   The analytical and FEA-predicted phase self-inductance are illustrated in Fig. 24, and the corresponding average inductances are listed in Table V. Besides, the slot leakage  component and air-gap component are obtained using  analytical model, as shown in Table VI. It is found that the self-inductance of the M-II machine is almost the double of the M-I machine. The slot leakage component is close to the air-gap component in the 48s/52p M-I MSTPM machine, which is 0.106 mH and 0.104 mH, respectively. However, in the 24s/26p M-II MSTPM machine, the air-gap component is quite higher than the slot leakage component, resulting from the large slot width. Besides, the analytical and FEA-calculated mutual-inductance are listed in Table VII. It is found that the 24s/26p M-II MSTPM machine has a more terrible saturation than the 48s/52p M-I MSTPM machine. Overall, a good agreement is achieved.

D. On-load Performance
Based on the back-EMF and inductance, the on-load performance is deduced. In Fig. 25, the analytical, FEA-predicted, and tested torque-current curves of two prototype machines are illustrated and compared. When armature current is less than 30 A, the analytical, FEA, and tested results of the 48s/52p M-I MSTPM machine are very close. When current increases, the error between analytical and FEA & tested results rises since the analytical method ignores the saturation.
As for the 24s/26p M-II MSTPM machine, the analytical, FEA, and tested torque-current curves are very similar throughout the whole current region.
Furthermore, the torque-speed curves of the two MSTPM machines are calculated as shown in Fig. 26. It is found the armature current raises the saturation in the 48s/52p M-I MSTPM machine, resulting in the difference between analytical results and FEA & tested results. Since the analytical model ignores the saturation, the analytical PM flux linkage is higher than the FEA and tested results, the corresponding average torque should be higher when the same current is injected. Further, considering the analytical and FEA inductances are very close, the analytical constant-power region is narrower than the FEA and experimental results. As for the 24s/26p M-II MSTPM machine, the armature current slightly influence the saturation so the analytical, FEA, and tested constant-torque region are similar. However, since the analytical inductance is higher than the FEA-predicted one, the analytical constant-power region is a bit wider. It is gratifying that the tested results are very close to the FEA-predicted results. Overall, good agreements have been achieved between the analytical, FEA, and tested results.

VI. CONCLUSION
In this paper, the analytical model of the MSTPM machines is proposed based on the AFM theory. Firstly, the MSTPM machines are divided into three key elements which are the open-circuit PM-MMF, the relative stator permeance, and the relative rotor permeance. The relative stator and rotor permeance are further deduced on the basis of RP model. Then, the general formula is proposed to calculate the back-EMF, as well as discover the relationship between spatial field harmonics and temporal back-EMF harmonics. The winding inductance model of the MSTPM machines for IWT applications are built considering the single-layer and double-layer winding configurations. Finally, the analytical model is verified by FEA-predicted and experimental results of two prototyped machines with different magnetization modes. It is believed that the proposed general model is an effective tool with considerable precision for the design and analysis of the MSTPM machines, especially for in-wheel traction applications.