Analytical Model of Air-Gap Field in Hybrid Excitation and Interior Permanent Magnet Machine for Electric Logistics Vehicles

With the increasing energy shortages and environmental degradation, electric logistics vehicle (ELVs) with energy conservation and environmental protection has become a research hotspot. The design of machine is the key to develop ELVs. Magnetic field analysis is the most critical issue since its accuracy affects the calculation of motor torque, loss, and other characteristics. To provide a calculation method for the field and performance analysis of the machine for ELVs, this paper presents an analytical model of air-gap field for hybrid excitation and interior permanent magnet machine. In the proposed model, it is taken into account for the shape of the stator and rotor teeth. The flux density on the rotor side is derived by equivalent magnetic circuit (EMC) with leakage magnetic flux. Taking the calculated flux density as one of the second boundary conditions, the air-gap field distribution is calculated by magnetic potential model with the eccentricity of the rotor. To verify the analytical method, we adopted the finite element method. The simulation results of the air-gap flux, back electromotive force, and cogging torque are in good agreement with the analytical results. Besides, applying the analytical model, the machine can be optimized for obtaining the optimal air-gap flux density distribution. The hybrid excitation machine with salient pole and interior magnets can provide a good flux density waveform. The study offers a helpful analytical method for design and optimization of the type of machine for ELVs.


I. INTRODUCTION
With the rapid development of e-commerce and express delivery, the challenges facing logistics vehicles are increasing [1], [2]. According to the European Commission, by 2030, the transportation of goods in central cities should produce no emissions at all. However, it accounts for 25% of urban carbon dioxide emissions, and the number of logistics vehicles is also increasing rapidly. A lot of research have been done on vehicle safety and comfort [3]- [6]. In the context of energy shortages and environmental degradation, the development of electric logistics vehicles (ELVs) is an effective solution to solve the problems and achieve sustainable development [7]- [9]. As the key component of the ELVs, The associate editor coordinating the review of this manuscript and approving it for publication was Zhiwu Li . hybrid excitation synchronous machines (HESMs) have the advantages of high power density of permanent magnet motor and controllable magnetic field of electric excitation motor, which can improve the efficiency of ELVs [10]- [12]. To improve the high efficiency of electric vehicles, the braking energy recovery and motor topology have always been the focus of many researchers [13]. Among various HESMs, parallel hybrid excitation synchronous machines (PHESMs) have some advantages over series hybrid excitation synchronous machines (SHESMs). The magnetic flux generated by the excitation windings does not pass through permanent magnets (PMs), reducing the magnetic resistance of the magnetic circuit and improving the machine efficiency [14]. The magnetomotive force (MMF) created by the armature windings forms a loop through the rotor core, avoiding the risk of PM demagnetization. By adjusting the excitation VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ current, and changing the value of the synthetic magnetic field in the air-gap, the machine can operate under different working conditions [15]- [17]. The no-load air-gap magnetic field distribution is the basis of the design performance index of HESMs, such as back electric motive force (back-EMF), torque-speed and other performance. For design and optimization, the finite element method (FEM) and analytical method are often used to calculate the magnetic field [18]- [21]. However, no author has applied the analytical model to calculate the magnetic field of hybrid excitation and interior magnets machine considering the shape of stator and rotor teeth and the leakage magnetic flux. The FEM can analyze the magnetic field of various types of machines considering the effects of saturation and nonlinearity on the magnetic field distribution [22], [23], but it takes lots of time and computer source, its application in machine optimization and performance analysis is limited to some extent [24]- [26]. In [27], an analytical method for calculating the air-gap field distribution of SHESM is described in detail. It shows a structure in which two excitation sources are located in the rotor, each PM is placed on the surface of the salient pole. In [28], based on subdomain model, it presents an analytical method to predict of magnetic field distribution of parallel hybrid excitation and spoke-type tangential magnets motor with considering the tooth tips and shape of polar pieces. Each PM is placed on between two adjacent salient poles respectively. References [30], [31] provide other topologies of HESMs with complicated geometric structure, in which the studies are only done by the FEM.
The interior permanent magnets machines have the advantages of high power density and high efficiency, and have been widely used. Especially multisegment or multilayer interior magnets machines, which have additional reluctance torque, the torque density is higher. Due to the complicated magnetic flux path and leakage magnetic flux, the performance of these interior magnets machines is generally analyzed by the FEM [32]- [35], and few scholars use the analytical method [36]- [39]. To deal with the problem that the FEM needs a lot of computation time in the design process, an approach using the lumped equivalent magnetic circuit (EMC) method to calculate the electromagnetic performance of the interior magnets machine is proposed [36], [37]. In [38], the flux density of a multi-layered and multi-segmented interior permanent magnet machine is calculated by combining the EMC and conformal mapping with slotting effect. The EMC can only estimate the radial component of the flux density. According to the author's knowledge, there is no research on the hybrid analytical method for the interior permanent magnets machine that considers the shape of the stator and rotor teeth and the leakage magnetic flux. However, many scholars adopted the hybrid analysis method combining EMC and Maxwell's equations in the spoke-type magnets motor. To obtain the boundary condition on the rotor side, different models are presented considering the irregular PMs shape, teeth saturation, and the nonlinearity of magnetic materials, respectively [40]- [42]. This paper presents an analytical model of air-gap field in hybrid excitation and interior permanent magnets machine for ELVs, in which the shape of stator and rotor teeth is taken into account. The model combines the EMC and Maxwell's equations. To calculate the boundary conditions, the EMC models under different paths of hybrid excitation and interior permanent magnets machine and a specific calculation method for main permeance are given. The air-gap magnetic field is obtained by Maxwell's equations considering the rotor eccentricity. The model is verified by the FEM. Moreover, utilizing the analytical model, the machine is optimized by design the suitable pole-arc coefficients and the rotor eccentricity. In Section II, the topologies and features of the machine for ELVs are given. In Section III, the EMC model and calculation method for main permeance are derived, the model of the magnetic field distribution with the rotor eccentricity is established. In Section IV, the analytical model is verified, and the main structural parameters are optimized to improve the fundamental and reduce total harmonic distortion (THD) of the air-gap flux density. In Section V, the conclusions are summarized.

II. TOPOLOGY OF THE MACHINE
The configuration of the hybrid excitation and interior permanent magnets machine is shown in Fig. 1. In the machine, the rotor is composed of PM1 and PM2 which are built in the outer middle of the pole shoe and the excitation winding placed on the pole body in a series manner, which can increase the sine of air-gap flux density waveform. The magnetic flux of each pole is provided by the PM1, PM2, and excitation winding, the flux produced by the excitation winding does not pass through magnets, which can improve the motor efficiency and avoid demagnetization of PMs. The magnetic barrier is arranged to reduce the leakage magnetic flux. The magnetic bridge is set to connect the slots for PMs and pole shoes for the PMs. To simplify, assuming that the stator and rotor permeability is infinite. The eddy current, end and conductivity effects are neglected.  Partial structures of the machine including the salient pole and stator tooth are shown as Fig. 2. To improve the sine of the air-gap flux density waveform, the outer edge of the rotor core is designed as the non-circular shape, as shown in Fig. 2 (a). Due to the irregular structure of the stator tooth, the segmentation calculation is adopted, the segmentation diagram can be seen in Fig. 2

III. ANALYTICAL MODEL FOR THE MACHINE
Due to the complicated rotor topology of the hybrid excitation and interior permanent magnets machine shown in Fig. 1, the analytical calculation model of magnetic field for the machine is derived in detail. Firstly, the EMC model is established to derive the flux density on the rotor outer surface. Then, the radial and tangential of the air-gap flux density are calculated by considering the effect of the non-arc pole shoes. Finally, on the basis of the air-gap flux density, the expressions of cogging torque and back-EMF are given.   magnetic pole can be divided into two components, which are generated by the excitation winding and PM, respectively. Setting that the permeability of the PM is equal to that of vacuum, the magnetic flux generated alone by the excitation winding is obtained, as shown in Fig. 3 (a). The main magnetic flux is through the magnetic bridge of PM left and right to linkage armature winding. There are three leakage flux lines, including the leakage flux through the additional air gap between two adjacent pole shoes, and the additional air gap between two adjacent pole body, the stator teeth close to the air-gap. And setting the value of the excitation current is zero, the magnetic flux generated alone by the PM1 and PM2 are obtained, as shown in Fig. 3 (b). Fig. 3 (b) indicates VOLUME 8, 2020 that the main magnetic flux created by PM1 and PM2 are both in series and in parallel. Similarly, there are three leakage flux lines, including the leakage flux through the magnetic bridge, the additional air gap between different poles, and the stator teeth close to air-gap.

A. MAGNETIC CIRCUIT MODEL
As shown in Fig. 3, the studied machine has three magnetic potential sources on each pole, including electric excitation winding, PM1, and PM2. When the electric excitation source acts alone, the EMC diagram of electromagnetic can be obtained, as shown in Fig. 4 (a).
In a similar way, the EMC diagram of the other two magnetic potential sources under separate action can be obtained in Fig. 4 (b) and Fig. 4 (c).
Applying the superposition principle in electrotechnics, the total magnetic circuit for hybrid excitation and interior magnet machine can be obtained, as shown in Fig. 5. Due to the three magnetic potential sources occupy the different length of the salient pole, each pole shoe is divided into three parts, and the air-gap cross-sectional area can be regarded as three segments. Fig. 5 shows that the flux through the first part of the air-gap is provided only by the excitation winding, and the flux through the second part of the air-gap is provided by the excitation winding and PM1. At the same time, the flux through the third part of the air-gap is provided by the electric excitation winding, PM1, and PM2.
According to Fig. 5, the expression of the EMC model can be derived as follows by applying the Kirchhoff's law of magnetic circuits: F e , F m1 , and F m2 are the MMF excited by excitation winding, PM1, and PM2 over one magnet pole, respectively, G m1 and G m2 are the magnetic permeance of PM1 and PM2, can be calculated by: where N is the number of excitation winding turns, I f is the excitation current, H r is the intensity of the permanent magnetic field, µ 0 is the permeability of the vacuum, µ m is the relative permeability of permanent magnet, w m1 and w m2 are the thickness of PM1 and PM2, b m1 and b m2 are the width of PM1 and PM2, L ef is the effective length of the stator and rotor. G Lm11 and G Lm12 are the leakage magnetic permeance of rotor core at top and bottom of PM1 over one magnet pole, G Lm21 and G Lm22 are the leakage magnetic permeance of rotor core at the top of PM2 and additional air gap at the bottom of PM2 over one magnet pole, respectively, G Lδ1 and G Lδ2 are the leakage magnetic permeance of the additional air gap between two adjacent pole shoes, can be calculated by: where µ r is the relative permeability of the rotor core, R r1 and R r2 are the outer and inner radius of pole shoe, l 0 is the length of the magnetic bridge between the PM and the outer side of the rotor, α pe is the pole-arc coefficient of the salient pole as shown in Fig. 6, θ r are the rotor teeth pitch. G tt and G ts are the tangential leakage magnetic permeance of stator tooth and slot, respectively, can be calculated by:  where h ti and b ti are the height and width of the ith region of the stator tooth, b so are the width of slot opening. G δe , G δm1 , and G δm2 are the magnetic permeance of the additional air gap between the rotor core and excitation winding, PM1, and PM2, respectively, G δi is the magnetic permeance of air-gap between the stator and rotor, can be calculated by: where b pb and h pb are the width and height of pole body, δ e , δ m1 and δ m2 are the additional air gap length between the rotor core and excitation winding, PM1, PM2. δ i is the length of air-gap between the stator and rotor, a i is the pole arc width in mechanical angle of the ith layer PMs. G ci is the magnetic permeance of the rotor core at the different position, can be calculated by: where S c and l c are the corresponding cross-sectional area and the length of the rotor core at different locations. According to the segmentation method shown in Fig. 2(b), the stator tooth is treated as the superposition of four different areas, including an approximate rectangle (I), two trapezoids (II and III) and a quadrangle (IV). Due to the departments of I, II, III and IV are series connection, and the total permeance of stator tooth G st can be calculated by: (19) x G st1 is the magnetic permeance in I region, and it can be calculated by: where R s2 is the outer radius of stator core, θ s is the stator teeth pitch. y G st2 is the magnetic permeance in II region, and it can be calculated by: z G st3 is the magnetic permeance in III region, and it can be calculated by: { G st4 is the magnetic permeance in IV region, and it can be calculated by: where R so is the outer radius of stator core, R s1 is the radius of stator slot bottom, R t is the fillet radius in IV region. G syi is the stator yoke magnetic permeance, and it can be calculated by: where n st is the amount of teeth across the stator.
The key permeability shown in Fig. 5 can be calculated by: Define A si as the effective magnetic flux area corresponding to the ith magnetic potential source, so, when ψ represents the angle with reference to d axis, the flux density of outer side of the rotor and the Fourier expansion on the calculated position can be expressed as [40]: VOLUME 8, 2020

B. MAGNETIC FIELD DISTRIBUTION MODEL
Based on Maxwell's equations, the magnetic potential distribution zξ can be expressed by the Laplace's equation: where r and α are the position of the calculated point, representing the radial length and the circumferential angle, respectively. The general solution of (28) in different regions can be expressed by separation of variables method as follows: where i, ii, and iii represent the regions of the slot, slotopening, and air-gap. A1, A2, A3, B3, C3, and D3 are the coefficients to be determined. α i is the angle of the ith slot, R ro is the maximum radius of the rotor surface. Under the same magnetic pole, the period of the flux density distribution is the same as that of the stator cogging. Equation (31) is simplified as: The boundary condition on the inside surface of the stator core can be calculated by: If R p is defined as the radius of the pole shoe surface, as shown in Fig. 6, the interfaces conditions on the outer surface of the pole shoe can be expressed by: where B ps_out represents the average flux density on the pole shoe outer surface. Combining (32) and (34), the radial magnetic flux density can be represented as: Substituting (33) into (32), it can be obtained by: Due to the salient pole structure of the machine, B ps_out is equal to B pr_out , the expression of interfaces conditions on the outer surface of the pole shoe by combining (32) and (35) can be derived and simplified by: In Fig. 6, h represents the eccentricity of the rotor, that is, the distance between the center of the rotor core and the salient pole, according to the geometric relationship in Fig. 6, h can be deduced by: (38) Selecting the point K outside the salient pole randomly, and considering its coordinate as (x, y), it can be obtained by: where λ is the distance between the calculated point K and the origin, α k is the angle between the line from point K to ordinate origin and the center axis of the salient pole. Submitting (38) to (39), the expression of the radius of the pole shoe surface can be obtained by: where ϕ is equal to 2π α p1 . According to (40), the partial derivative of the radius of the pole shoe surface is calculated by: Furthermore, the outside edge of the pole shoe is regarded as a parabola opening toward the origin of the coordinates, as shown in Fig. 6, the distance λ can be calculated by: In (42), is the function expression of the proposed parabola, the variable can be expressed by derivation and simplification: Substituting (43) to (41), and it can be obtained by: Therefore, according to (36), (37) and (41), the coefficient A 3 can be obtained by: Based on (32), (38) and (44), the radial and circumferential of the air-gap flux density are calculated by: Thus, combining (46), (47) with the Maxwell tensor method, by integration, cogging torque of the hybrid excitation machine is derived as: The back-EMF of hybrid excitation machine is induced by the flux linkage in the winding which changes with time, it can be expressed as [17]: where σ represents the phase A, B, and C of armature winding, a represents the number of parallel branches, N c represents the turns per coil, α n represents the angle of the slot where the coil is located. The electromagnetic torque can be obtained by the principle of virtual displacement: where w represents the rotor speed in rad/s, E A , I A , E B , I B , E C , and I C represent the back-EMFs and currents of phase A, B and C, respectively.

IV. FEA VALIDATION AND PERFORMANCE ANALYSIS
In this section, the calculated results of the radial and tangential of air-gap flux density, Back-EMF, and the cogging torque are compared with simulation results of that, the correctness of analytical model is verified. Based on the proposed analytical model, an optimization is performed to maximize the amplitude of the fundamental air-gap flux density and minimize THD of the air-gap flux density.

A. FEA VALIDATION
To verify the rationality and effectiveness of the analytical model, the 8-pole/48-slot hybrid excitation and interior permanent magnets machine is chosen as the simulation prototype. The main calculation and simulation parameters of the machine are listed in Table. 1. The parameters of the rotor are listed in Table. 2. The simulation model is established by the simulation software, in which all magnets is set as NdFeB.
The stator and rotor adopt linear materials to approximate the infinite permeability in the assumptions in the simulation model. In the section, the simulation results and calculated results of the machine, including the air-gap flux, back-EMF and cogging torque are compared, respectively.  The air-gap flux density of the hybrid excitation and interior permanent magnets machine is drawn in Fig. 7. Selecting the center position of the air-gap, the radial and tangential component of the air-gap field are calculated. It can be seen that for the eccentric salient pole structure of the hybrid excitation and interior permanent magnets machine, the curve of the analytical results and that of the simulation results have VOLUME 8, 2020 a high degree of agreement, which shows that the analytical model can calculate the radial and tangential of air-gap flux density. In addition, the analytical results in the stator slot and the salient pole are similar to the simulation results, indicating that the analysis model can consider the influence of the shape of the stator and rotor teeth on the air-gap flux density. It is worth noting that the sine of the air-gap flux density waveform is relatively high.   8 shows that the Back-EMF curves of single phase under no-load condition. The simulation curve of the Back-EMF is very close to the calculation curve of that, which proves the analytical model can be used to obtain the Back-EMF of the studied machine under no-load condition. Besides, the curves shown in Fig. 8 are close to the sinusoidal waveform, which indicates that the structure of the provided salient pole rotor can enhance the output quality of the machine. Fig. 9 shows that cogging torque curves obtained by analytical calculation and simulation, respectively. It can be seen that the cogging torque waveform predicted by the analytical model has a small difference with the simulation result, which further proves the feasibility and practicability of the analytical method. Due to the analytical model of magnetic field proposed in this paper takes into account the influence of stator slotting and the non-circular outer edge of the rotor,  the cogging torque model obtained by the calculation method is also affected by the cogging effect. Table. 3 shows the comparisons of the root mean square (RMS) of radial and tangential air-gap flux density, back-EMF and cogging torque of the machine. It can be seen that using the analytical model and FEA, the RMS of radial air-gap flux density of the machine is about 0.5 T, and the cogging torque is about 500 mNm, the back-EMF is about 21 V, which indicates that the RMS of calculated by the analytical model are close to those of FEA.

B. PERFORMANCE ANALYSIS
The pole-arc coefficient corresponding to each excitation source of a single magnetic pole is a crucial factor affecting the amplitude of the fundamental of air-gap field and the distribution of the harmonic of air-gap field. Due to the space of pole shoe of the studied machine is limited, in order to maximize the electromagnet field utilization, the pole-arc coefficient of the pole shoe through the electric excitation source α pe is selected as 0.91. Moreover, the non-circular structure of the rotor surface is adopted in the machine. It is quite obvious that the eccentricity is also a crucial factor which can affect the distribution of air-gap field. h is defined as the eccentricity of the rotor, α p1 and α p2 are defined as the pole-arc coefficients of the PM1 and PM2. Therefore, the influence of the arrangement form of the pole shoe and PM1, PM2 is researched by using the proposed analytical model, that is, the different combination of h, α p1 and α p2 are analyzed for the optimum distribution of air-gap field, as shown in Fig. 10-12. Fig. 10 shows the amplitude of the fundamental air-gap flux density under the different eccentricity. In Fig. 10 (a), it can be seen clearly that when α pe remains unchanged, the fundamental amplitude of the air-gap flux density enhances as α p1 and α p2 increase, which is caused by the increase in volume of PMs. Besides, compared with α p2 , α p1 has a greater influence on the fundamental of air-gap flux density. Further, it can be seen that compared with α p1 and α p2 , the eccentricity has the greatest effect on the fundamental amplitude of the air-gap flux density, as shown in Fig. 10 (b). When h is set to 0, 5, 10, 15, and 20, respectively, the fundamental amplitude of the air-gap flux density increases slowly at first, and then more quickly.
Keeping α pe constant and changing α p1 , α p2 , and h, the variation of THD of the air-gap flux density will be more complicated. Therefore, The THD with different eccentricity h is analyzed, in which the pole-arc coefficients of PM1 and PM2 are set within a standard range, it is because of the need to ensure the maximization of the sine flux waveform. From Fig. 11, we can see that h has the most significant influence on the THD, whereas α p2 has the least influence on the THD. Furthermore, with the increasing of h, the THD decreases at first and then increases gradually. When h is from 0 to 10, THD decreases, the minimum value of THD decreases from 0.2 to 0.17. When h is from 10 to 15, THD increases, the minimum and maximum of THD increases from 0.17 to 0.19, and from 0.29 to 0.34, respectively, but the increasing of the THD is smaller. When h is from 15 to 20, THD still increases, however, compared with the previous one, the scope of the increase is bigger. Besides, All the THD with different eccentricity shown in Fig. 11 is less than 0.3. When h is 10 or 15, the area with the THD below 20% is the largest, which can be seen in Fig. 11 (c) and Fig. 11 (d). When h is 20, the area with the THD above 31% is the widest, which can be seen in Fig. 11 (e).
By combining Fig. 10 (b) and Fig. 11 (e), it is obvious that with the increasing of h, the fundamental air-gap flux density always keeps increasing, but when h exceeds a certain value,  the THD also increases. Therefore, both the fundamental and THD of air-gap flux density should be taken into account to design the eccentricity of the rotor. Aiming at the minimum THD, the optimal parameters h, α p1 , and α p2 can be obtained. The optimal combination of α p1 and α p2 with different h is given, as shown in Fig. 12. Similar to the results obtained in Fig. 11, when h is from 10 to 15, the THD is the smallest. α p1 and α p2 are 0.65 and 0.25, respectively, and h is 15, minimum THD is obtained. When the eccentricity of the rotor, the pole-arc coefficients of the PM1and PM2 are 15, 0.65 and 0.25, the radial air-gap flux density distribution is given, as shown in Fig.13. Comparing with the initial radial air-gap flux density curve in Fig.7, the air-gap flux density waveform of the optimized machine is closer to a sine curve, and the analytical result is in good agreement with the simulation result.

V. CONCLUSION
In this paper, aiming at the magnetic field analysis of the hybrid excitation and interior permanent magnets machine for ELVs, the analytical method is proposed for predicting the air-gap field distribution. In the analytical model, the EMC method and magnetic potential analysis method are combined, and the shape of the stator and rotor teeth is considered. To obtain the boundary conditions, the EMC model considering the stator teeth shape and the leakage flux is established for calculating the air-gap flux density on the rotor side, which can also provide theoretical foundation for reducing the magnetic leakage. Applying Maxwell's equation, the magnetic field model is obtained by accounting for the eccentricity of the rotor.
The analytical method is verified by using the simulation model. The result shows that the analytical model can be adopted to calculate the electromagnetic performance of this type of machine, including the air-gap field, back-EMF and cogging torque. By utilizing the analytical model, the pole-arc coefficients of PM1, PM2, and the rotor eccentricity are optimized to get maximum fundamental and minimum THD of air-gap flux density. Moreover, compared with the former, the eccentricity has a greater influence on the air-gap field distribution of the studied machine. For the hybrid excitation and interior permanent magnets machine, under the condition of ensuring the maximization of electromagnetic utilization, the distribution of the air-gap field can be improved by changing the pole-arc coefficients of PM1 and the rotor eccentricity, thus improving the operational quality of the machine. In addition, the machine can improve the quality of the air-gap flux waveform to make the THD of that below 30%.
There is still much work to be done on the analytical modeling of the hybrid excitation and interior permanent magnets machine for ELVs. In the future work, the application of the method to other machines of this type will be involved. In order to get more precise results, it is necessary to consider the nonlinear characteristics of the materials of stator and rotor in the machine. In addition, the efficiency and control strategy of the machine should also be further studied.  He is currently working on research and development of vehicle's motors and electric vehicles with the Transportation and Vehicle Engineering School, SUT. He holds 32 authorized patents. He has published 22 articles and two monographs. He was a recipient of the Second Award of National Technical Invention and five items of the second prizes at provincial and ministerial level.

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GUODONG LIU received the B.S. degree in mechanical design, manufacturing and automation from Ludong University, Yantai, China, in 2019. He is currently pursuing the M.S. degree with the Transportation and Vehicle Engineering School, Shandong University of Technology, Zibo, China.
His research interests include vehicle engineering, automotive electronics, design and manufacture of motor, and new energy electric vehicle technology. VOLUME 8, 2020