3-D Analytical Model of Armature Reaction Field of IPMSM With Multi-Segmented Skewed Poles and Multi-Layered Flat Wire Winding Considering Current Harmonics

A 3-D analytical model of armature reaction field (ARF) of interior permanent magnet synchronous motor (IPMSM) with multi-segmented skewed poles (MSP) and multi-layered flat wire winding (MFWW) considering current harmonics is proposed in this paper. Firstly, based on the MFWW distribution analysis and the coil magnetic field vector superposition method, the ideal ARF functions of the MFWW considering low-order current harmonics and high-order sideband current harmonics are derived. Secondly, 3-D relative permeance functions caused by the stator slotting effect and the multi-segmented rotor magnetic barriers are presented. Thirdly, 3-D analytical expressions of ARF are obtained by multiplying the ideal ARF functions and the 3-D relative permeance functions. Additionally, the effects of low-order current harmonics, high order sideband current harmonics, MSP, and stator slotting on the spatial orders and frequencies of ARF are analyzed. Finally, the proposed model is verified by finite element method (FEM) and experiment.


I. INTRODUCTION
Currently, interior permanent magnet synchronous motor (IPMSM) has been widely equipped in electric vehicles, due to its several advantages compared with surface permanent magnet synchronous motor (SPMSM), such as high flux-weakening capability, high torque output owing to the reluctance torque, and robust structure because of permanent magnets embedded in the rotor [1]. The multi-segmented skewed poles (MSP) and the multi-layered flat wire winding (MFWW) further enhance the performance of IPMSM. Firstly, step skewing of the rotor effectively reduces its The associate editor coordinating the review of this manuscript and approving it for publication was Kan Liu . cogging torque and torque ripple [2], [3]. Secondly, the MFWW, namely hairpin winding, achieves a higher fill factor, shorter end windings, higher power and torque density and lower copper loss [4], [5]. Furthermore, because of a better contact between conductors and between winding and iron core, a better cooling performance compared with traditional round wire winding is gained [6]. Therefore, IPMSM with MSP and MFWW has a promising prospect. However, current harmonics caused by nonsinusoidal back electromotive force (BEMF) and pulse width modulation (PWM) significantly influence the armature reaction field (ARF) of IPMSM, and further affect inductance, iron loss, torque, vibration and noise [7]- [9]. Consequently, an accurate prediction of ARF of IPMSM with MSP and MFWW considering current harmonics is vital to achieve a better performance [10].
Nowadays, finite element method (FEM) and analytical method are two main approaches to calculate ARF of IPMSM. The major advantage of FEM is its ability to consider a complex rotor structure [11] and magnetic saturation. However, it is time consuming [12]- [15], especially for 3-D models, which are required for IPMSM with MSP. Compared with FEM, analytical method can explicitly reveal the sources of spatial and temporal harmonics [10], [15], and effectively guides IPMSM design. There are three analytical methods to predict ARF of IPMSM, i.e., magneticequivalent-circuit method, subdomain method and magneticpotential-and-permeance method: 1) In magnetic-equivalent-circuit method, magnetic resistances and magnetomotive forces of different areas need to be calculated at first. Then the magnetic circuit is compared to an electric circuit and finally the air-gap magnetic field density is obtained [14], [16], [17]. The effects of the complex rotor structures and the magnetic saturation on the ARF of IPMSM can be considered by magnetic-equivalent-circuit method. However, its accuracy is sensitive to the density of magnetic circuit nodes, and it can only predict the radial component of air-gap flux density but not the tangential component [18].
2) For subdomain method, IPMSM is usually divided into several areas, such as permanent magnets, slots, air-gap, and stator/rotor iron core. Then the governing partial differential equations based on Maxwell's equations are presented and solved. Finally, based on boundary conditions of different areas, coefficients of the general solution are determined and magnetic field distribution is obtained [9]- [11]. However, if permanent magnets are complexly embedded into the rotor, more subdomains are required for approximation. It complicates boundary conditions and makes equation solutions extremely complicated.
3) For magnetic-potential-and-permeance method, the ideal ARF without slotting is initially calculated by using winding coefficients [8], [15], [19] or the vector-addition method [20], [21]. Then the ARF with slotting can be calculated by multiplying the ideal ARF without slotting and the relative permeance function. Compared with magnetic-equivalent-circuit method and subdomain method, magnetic-potential-and-permeance method has simpler calculation processes, whereas its accuracy greatly relies on its relative permeance functions. There are several approaches to calculate the relative permeance functions. In [22], a subdomain method is adopted to calculate the relative permeance function of stator slots. In [23], a 2-D relative permeance can only predict the radial component of flux density but not the tangential component. In [12] and [21], the complex relative permeance based on conformal mapping can predict both kinds of flux density. However, the magnet barrier structure of step skewing rotors leads to the axial distribution variation of ARF, which is not considered in the references above. Moreover, few references take into account the effects of current harmonics on the ARF of IPMSM with MSP and MFWW.
In summary, a 3-D analytical model of ARF of IPMSM with MSP and MFWW considering current harmonics is required in order to meet the requirement of fast and accurate prediction. The remainder of this study is organized as follows. In Section II, based on the MFWW distribution analysis and the coil magnetic field vector superposition method, the ideal ARF functions of the MFWW considering loworder current harmonics and high-order sideband current harmonics are derived. Then, 3-D relative permeance functions caused by the stator slotting effect and the multi-segmented rotor magnetic barriers are presented. Finally, 3-D analytical expressions of ARF are obtained by multiplying the ideal ARF functions and the 3-D relative permeance functions. Additionally, the effects of low-order current harmonics, high order sideband current harmonics, MSP, and stator slotting on the spatial orders and frequencies of ARF are analyzed. In Section III, the proposed analytical model is directly verified by FEM and indirectly verified by measuring electromagnetic torque of an IPMSM.

II. ANALYTICAL MODELING
In order to illustrate the 3-D analytical modeling method of ARF, an IPMSM with 8 poles, 48 slots, 6 segmented skewed poles, and 4 layered flat wire windings is taken as a case. Its parameters are listed in Table 1. A. LOW-ORDER CURRENT HARMONICS AND HIGH-ORDER SIDEBAND CURRENT HARMONICS Generally, current harmonics can be divided into two categories: low-order current harmonics and high-order current harmonics [7]. The former are mainly derived from non-sinusoidal BEMF, which comes from non-sinusoidal VOLUME 8, 2020 open-circuit air gap field (OCAGF), while the latter are mainly caused by PWM [7], [24], [25]. Besides the fundamental current, there are plenty of low-order current harmonics and high-order sideband current harmonics in the phase currents of IPMSM. Therefore, three phase currents can be expressed: I n,k,1 sin 2π a n f s + b n,k f c + ψ n,k,1 where, I 1 , f c , and ψ 1 are amplitude, frequency, phase angle of the fundamental current, respectively, I h and ψ h are amplitude and phase angle of the h-order current harmonic, respectively, I n,k,1 and I n,k,2 are amplitudes of the high-order sideband current harmonics, ψ n,k,1 and ψ n,k,2 are phase angles of the high-order sideband current harmonics, a n and b n,k are coefficients of the high-order sideband current harmonics, f s is the switching frequency of PWM. According to (1)-(3), the main frequencies of the loworder current harmonics and the high-order sideband current harmonics are hf c and a n f s ± b n,k f c respectively.

B. IDEAL ARF OF MFWW
The cylindrical coordinate system O-rαz is established by taking the circle center of the end face of the stator core as the origin O, and α = 0 pointing to the center of the slot opening of Slot 1, shown in Fig. 1.
In order to consider low-order current harmonics and high-order current harmonics, the radial and tangential ARF by the l th coil can be expressed as Fourier  series: where, I l_i is the current in the l th coil, i represents Phase A, Phase B, or Phase C, I l_i ∈ {I A (t) , I B (t) , I C (t), q is the winding distribution order, B qr and B qt are the ampitudes of the q-order radial and tangential flux density when unit amplitude current is passed through the l th coil, respectively, α l is the angle of the center of the l th coil. Multiple coils or groups of coils constitute windings based on a certain distribution rule. The ARF of IPMSM can be equivalent to the superposition of magnetic fields of these multiple coils or groups of coils [26], [27]. Therefore, investigation of the winding distribution rule is the premise of analytical calculation of ARF of IPMSM. In this paper, because the greatest common divisor of the pole pair number and the slot number is equal to N = GCD(p, Q s ) = 4, the unit motors number is N = 4. Therefore, the winding distribution of the 4 layered flat wires repeats four times in the stator slots, shown in Table 2.
Based on the right hand rule, the coil magnetic field vector star chart of MFWW of IPMSM can be derived further, shown in Fig. 2. If upper and lower case letters represent forward and backward windings respectively, the magnetic field vector of each unit motor can be expressed as AAccBBaaCCbb.
Therefore, the ideal ARF of MFWW can be expressed as the superposition of ARFs of all the coils: where, Q is the coil number.

C. 3-D RELATIVE PERMEANCE FUNCTIONS
The stator slotting and the magnetic barriers of MSP rotor usually change the magnetic path reluctance and further affect the ARF. In order to consider the stator slotting effects, an effective 2-D complex relative air-gap permeance function has been derived in [12]. In order to consider the influence of axial direction, it is expanded into 3-D complex relative permeance function in this paper: where, λ sa and λ sb are the real part and the imaginary part of the complex relative air-gap permeance function considering the stator slotting effects, respectively, j is the imaginary symbol, λ saµ and λ sbµ are coefficients of Fourier series of the real part and the imaginary part, respectively, λ sa0 is constant of the real part, N µ is the highest order of Fourier series. In order to consider the permeance fluctuations caused by magnetic barriers of the rotor, numerical-analytical coupling methods, in which a fictitious magnet with a constant magnetomotive force is adopted to excite a slotless finite element model without magnetic barriers and a slotless finite element model with magnetic barriers, were developed in [8] and [28]. However, different fictitious magnet might lead to a different non-homogeneously saturated rotor [15]. In other words, the calculation methods of relative permeance function caused by magnetic barriers of the rotor in [8] and [28] depend on the parameter selection of the fictitious magnet. Therefore, a different method, in which a slotting finite element model without magnetic barriers and a slotting finite element model with magnetic barriers are excited by the ARF caused by MFWW at arbitrary time, is proposed to predict the relative permeance function caused by magnetic barriers of the rotor in this paper, shown in Fig. 3. The real part and the imaginary part of relative permeance caused by magnetic barriers of the rotor can be expressed [8], [28]: where, B ra_unr and B ra_unt are the radial component and the tangential component of air-gap magnetic field without magnetic barrier, respectively, B ra_r (r, α) and B ra_t (r, α) are the radial component and the tangential component of air-gap magnetic field with magnetic barrier, respectively. The rotor rotates at angular speed ω r and passes through time t. Due to periodicity, (9) and (10) can be expressed as Fourier series [8], [28]: where, λ rak and λ rbk are the real part and the imaginary part of the complex relative air-gap permeance function considering the magnetic barriers of the rotor, respectively, λ ra0 is constant of the real part, N k is the highest order of Fourier series. However, the 2-D analytical methods in [8] and [28] are not applicable to the relative permeance calculation caused by the magnetic barriers of MSP. As a consequence, a 3-D relative permeance of magnetic barriers of the rotor is presented in VOLUME 8, 2020 this paper. The axial length of the rotor is L, and the rotor is evenly divided into N 0 segments. The skew angle corresponding to the i th Segment is ψ i , shown in Fig. 4. For any point P(r, α, z) in space, its skew angle should be ψ Nz / L +1 , where ' ' represents the rounding-down function. Therefore, the relative permeance of the magnetic barriers of the MSP rotor can be expressed:

D. ARF OF IPMSM WITH MSP AND MFWW
The stator slotting and the magnetic barriers of the MSP rotor can change the relative permeance and affect the ARF of IPMSM with MSP and MFWW. Therefore, the radial and tangential components of ARF can be expressed [12]: B ar (r, α, z, t) = B less ar · λ ra + B less at · λ rb · λ sa + B less at · λ ra − B less ar · λ rb · λ sb (13) B at (r, α, z, t) = B less at · λ ra − B less ar · λ rb · λ sa − B less ar · λ ra + B less at · λ rb · λ sb (14) By Substituting (1)- (8) and (12) into (13) and (14), a 3-D analytical model of ARF of IPMSM with MSP and MFWW considering current harmonics can be obtained: B at (r, α, z, t) Based on (15) and (16), the spatial order and frequency features of ARF can be derived, in Table 3. The low-order current harmonics and the high-order sideband current harmonics do not change the spatial order q of the ideal ARF, but the frequency components hf c and a n f s ± b n,k f c in ARF are added respectively. On the one hand, spatial orders 0 and µQ s , and the frequency 0 Hz in relative permeance appear due to the stator slotting. On the other hand, the magnetic barriers of the rotor generate the relative permeance with the spatial orders 0 and 2kp, and the frequencies 0 Hz and 2kf c Hz. As a result, these factors significantly influence the spatial order and frequency features of ARF. Its main spatial orders are q, q ± µQ s , q ± 2kp, and q ± 2kp ± µQ s . Its prime frequencies are f c , hf c , a n f s ± b n,k f c , (2k ± 1) f c , (2k ± h) f c , and a n f s ± b n,k + 2k f c .

III. VALIDATION OF PROPOSED MODEL A. COMPARISON BETWEEN ANALYTICAL AND FEM RESULTS OF ARF
In order to verify the proposed analytical model, the analytical results and the FEM results of ARF at the rated  speed 5100 rpm and the load torque 100 N·m are compared in this chapter. Temperature of the FEM model is setted to 90 degrees Celsius.
Three phase currents were measured through experiment, shown in Fig. 5. A large number of harmonic components result in unsmooth time history waveforms of phase currents. component of phase currents, but there are plenty of low-order current harmonics with frequencies hf c and high-order sideband current harmonics with frequencies a n f s ± b n,k f c .
The experimental phase currents are inputted into the proposed analytical model and the finite element model and then the ARF can be obtained, shown in Fig. 6, Fig. 7, and Fig.8. It can be concluded that the analytical results agree well with the FEM results.
The time history and the amplitude-frequency characteristics of ARF are demonstrated in Fig. 6. The fundamental current with the frequency f c generates the fundamental ARF with the same frequency. Meanwhile, the low-order current harmonics with the frequencies hf c and the high-order sideband current harmonics with the frequencies a n f s ± b n,k f c  respectively generate the low-order and the high-order ARF harmonics with the same frequencies. Additionally, the magnetic barriers of the rotor cause new ARF harmonics. Under the action of the magnetic barriers and the current harmonics, ARF harmonics with new frequencies (2k ± 1) f c , (2k ± h) f c , a n f s ± b n,k + 2k f c occur.
The spatial distribution and order characteristics of ARF are displayed in Fig. 7. The spatial order of the ideal ARF is q. In addition, the stator slotting and the magnetic barriers of the rotor respectively generate new spatial orders µQ s and 2kp. Therefore, the main spatial orders of the ARF of IPMSM are q, q ± µQ s , q ± 2kp, and q ± 2kp ± µQ s . The analytical results and the FEM results of the 3-D ARF of IPMSM with MSP and MFWW are exhibited in Fig. 8.
Although the MSP structure does not change the frequency characteristics and the spatial orders of ARF, it leads to the variation of the relative permeance along the axial direction because of the position modification of the magnetic barriers of the rotor and further causes the axial distribution variation of ARF.
The calculation efficiency of the analytical model and the FEM are compared further. In this case, a Dell Precision T5610 CTO Base with 16 cores, 96 G memory, and 3 TB hard disk is adopted in both calculation methods. The time and space sample points numbers of two methods are the same. They are 541 and 1001, respectively. The calculation time of the analytical model and the FEM are about 40 minutes and 40 hours, respectively. The former is only 1/60 of the latter. It can be concluded that the proposed analytical model in this paper is of high efficiency.

B. INDIRECT VALIDATION WITH ELECTROMAGNETIC TORQUE MEASUREMENT
Since ARF is difficult to be measured, measurement of the electromagnetic torque is adopted to indirectly verified the proposed model. Test bench for measurement is AVL Dyospirit 200/3.2-16 and the motor runs at the rated speed 5100 rpm and the load torque 100 N·m. The radial and tangential components of air gap magnetic field can be expressed by superposition method: B r (r, α, z, t) = B ar (r, α, z, t) + B mr (r, α, z, t) where, B mr (r, α, z, t) and B mt (r, α, z, t) are the radial and tangential components of OCAGF, respectively. They can be obtained by a 3-D Analytical Model of OCAGF of IPMSM [29]. By substituting (15) and (16) into (17) and (18), respectively, the air gap magnetic field is obtained and the electromagnetic torque can be expressed as [30]: where, B r (α, z, t) and B t (α, z, t) are the radial and tangential components of air gap magnetic field in the middle of the airgap, respectively, r is the radius of the middle of airgap, µ 0 is the permeability in the airgap. By substituting (17) and (18) into (19), electromagnetic torque is obtained. It is compared with experiment and FEM results, shown in Fig. 9. It can be concluded that the experiment results have less fluctuation than other results. FEM results are more approximate to the experiment results compared with analytical results. The differences between experiment results and two predicted results are 1.60% and 8.24%, which are not big in general. Experiment indirectly verifies the proposed model.
It needs to be explained that the experiment results fluctuates little due to the low sampling frequency of the sensor. In addition, because of the superposition method used in (17) and (18), part of the magnetic saturation in iron core is ignored, and then the resultant air gap magnetic field is overestimated. When this resultant air gap magnetic field is used in torque calculating, the calculated torque will be larger than experiment as it is shown in Fig. 9.

IV. CONCLUSION
In this paper, a 3-D analytical model of ARF of IPMSM with MSP and MFWW considering the low-order current harmonics and the high-order sideband current harmonics is proposed. The main conclusions are as follows: 1) The low-order current harmonics with frequencies hf c and the high-order sideband current harmonics with frequencies a n f s ± b n,k f c generate the low-order and high-order ARF harmonics with the same frequencies, respectively. Additionally, the magnetic barriers of the rotor together with the current harmonics generate new ARF harmonics with frequencies (2k ± 1) f c , (2k ± h) f c , and a n f s ± b n,k + 2k f c .
2) The stator slotting and the magnetic barriers of the rotor respectively bring in new spatial orders µQ s and 2kp of ARF.
The main spatial orders of the ARF of IPMSM are q, q±µQ s , q ± 2kp, and q ± 2kp ± µQ s .
3) Although the MSP structure does not change the frequency characteristics and the spatial orders of ARF, it leads to the variation of the relative permeance along the axial direction because of the position modification of the magnetic barriers of the rotor and further causes the axial distribution variation of ARF.
The proposed model combines two permeance functions considering stator slotting and rotor magnetic barriers and expands them from 2-D to 3-D. It can calculate ARF of IPMSM with different slot-pole combinations, rotor structures. However, its accuracy relies on precise calculation of 3-D relative permeance functions.  His research interests include vehicle vibration and noise control, and the vibration and noise of electrical machines.