Issue 
Mechanics & Industry
Volume 24, 2023



Article Number  18  
Number of page(s)  17  
DOI  https://doi.org/10.1051/meca/2023005  
Published online  24 May 2023 
Regular Article
Research and control of a new dualmodulation magnetic gear compound motor for electric vehicles based on a mathematical model and FEA cosimulation
^{1}
School of Mechatronics and Vehicle Engineering, East China Jiaotong University, Nanchang 330000, China
^{2}
Key Laboratory of Conveyance and Equipment, Ministry of Education, East China Jiaotong University, Nanchang 330013, Jiangxi, China
^{*} email: mewjg@ecjtu.edu.cn
Received:
19
December
2021
Accepted:
27
January
2023
According to the development of the electric vehicle motor drives, a magnetic gear compound motor with small size, lightweight, noncontact, and highpower has a good development prospect in the new energy electric vehicle industry. A new dualmodulation magnetic gear compound motor (DMFMCM) with high torque, low torque ripple, and high mechanical strength is proposed in the paper. The topology and driving principle of DMFMCM are analyzed in this paper, and the finite elements analysis (FEA) is used to compare and analyze the DMFMCM and the conventional magnetic gear compound motor (CMGCM). Furthermore, the mathematical model of DMFMCM is established to achieve the decoupling of the threephase voltage or current, and the mathematical model and FEA cosimulation is established to ensure the accuracy of the mathematical model. Finally, according to the principle of SVPWM controlled by i_{d} = 0, the paper simulates a PIadjusted threephase DMFMCM control system model. The result shows that the DMFMCM controlled by SVPWM has high stability, strong antiinterference ability and good speed regulation performance, thus meeting the development of electric vehicles.
Key words: Magnetic gear compound motor / dualmodulation / simulation / SVPWM
© J.G. Wang et al., Published by EDP Sciences 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
In order to respond to the sustainable social development trend, electric vehicles have gradually replaced traditional fuel vehicles in the automotive industry. Electric vehicles convert electrical energy into mechanical energy by driving motors, which are environmentally friendly, stable, efficient and more convenient to operate than traditional fuel vehicles [1–3]. In order to respond to the trend of sustainable social development, the drive motor should have the characteristics such as low speed, high torque, high stability, and good safety and comfort [4–5].
The power system of electric vehicles is generally composed of a drive motor and a gear transmission system. The driving motors of electric vehicles mainly include DC motors, induction motors, switched reluctance motors, and permanent magnet motors [6]. The principle of the DC motor is simple, and the torque control is simple. However, its volume is usually large, the energy density and power density are low, and the commutation device has a life limit. Therefore, there are certain restrictions on the use of electric vehicles [7]. The induction motor has a simple structure and high reliability. However, with its electrically exciting system, the copper and eddy current losses can lead to lower efficiency and higher control costs. [8]. Switched reluctance motors have the characteristics of a simple rotor structure, high mechanical strength and reliability. However, the switched reluctance motor has high torque fluctuations at low speeds, which will seriously affect the comfort of electric vehicles during starting, braking and lowspeed driving [9]. In contrast, permanent magnet synchronous motors use permanent magnets instead of excitation winding. Permanent magnet synchronous motors reduce the copper and eddy current losses caused by the field current, so the motor efficiency is higher [10]. The permanent magnet motor is combined with the magnetic gear to form a compound motor with integrated drive and transmission. The transmission system of the magnetic gear compound motor has a transmission ratio, which can realize variable speed transmission. In addition, the magnetic gear compound motor has the advantages of small size, lightweight, low friction loss, convenient control, and high efficiency, which can meet the driving performance requirements of electric vehicles.
The magnetic gear compound motor, a new directdrive motor with low speed and high torque characteristics, was proposed by Professor K.T. Chau of the University of Hong Kong in 2009. This compound motor makes the magnetic gear and the permanent magnet motor share a rotor to form a compound structure, which improves the efficiency and power density of the device [11]. However, the compound motor has three air gaps, which are not conducive to control, and the rotor surface attaches to three layers of permanent magnets, which will fall off when the motor rotates at high speed. In 2014, Fan Ying from Southeast University in China proposed a unipolar permanent magnet arrangement magnetic gear compound motor with a selfdeceleration function [12]. This structure adds a modulation ring between the rotor and the stator to achieve variablespeed transmission. In addition, the twolayer air gap reduces the magnetic leakage phenomenon, which improves motor torque. In 2017, K. Atallah from the University of Sheffield in the UK proposed a pseudodirect drive magnetic gear compound motor [13]. The motor adopts an outer stator structure and has permanent magnets attached to the stator teeth, so there is one less air gap than the traditional magnetic gear compound motor. According to the analysis and experiment, the feasibility of the variable winding structure is verified, and the magnetic leakage phenomenon is reduced. In the same year, S.X. Niu from the Hong Kong Polytechnic University proposed a doublepole magnetic gear compound motor [14]. The outer rotor and the modulation ring have equal permanent magnets, so the two layers of permanent magnets are combined with iron sheets to form a modulation ring with modulation. Compared with the singlepole magnetic gear compound motor, this structure has a higher torque density and is more suitable for directdrive systems with low speed and high torque. In addition, Hang Zhao from the City University of Hong Kong proposed a dualmodulator magnetic gear reduction motor in 2020 [15]. The motor structure adds a modulator outside the spoketype permanent magnet motor, which makes it have a magnetic flux focusing effect and increases the flux density on the air gap to increase the torque. In summary, it can be found that the magnetic gear compound motor will strive to simplify the mechanical structure of the compound motor, reduce the number of air gap layers, and increase the torque density and reduce torque ripple.
In order to improve the performance of the drive motor, it is not only necessary to optimize the electromagnetic structure of the motor but also to realize the control algorithm. The controller is an important part of the electric vehicle, and the control algorithm is its core elements. In recent years, motor control technology is keeping maturing, providing reliable technical support for magnetic gear compound motors. The magnetic gear compound motor is a multivariable and strongly coupled control object. Therefore, decoupling its control parameters is the most effective way to improve the accuracy of the control algorithm, and vector control and direct torque control are the most common decoupling control strategies [16–18]. Although direct torque control has a low degree of model dependence, it has a longer control cycle and a smaller motor inductance. During startup and load changes, the torque ripple of the motor is relatively large. In addition, it is difficult to accurately observe the flux linkage and torque when the motor is at low speeds, and it is also hard to achieve highperformance control of the motor torque and flux linkage [19]. Vector control is also called fieldoriented control, and the control effect is based on the accurate mathematical model of the controlled object. By decoupling the stator current, the vector control decomposes the current into electromagnetic torque current and excitation current, and then the two currents are independently controlled. The vector control system can ensure the torque of the actual load object, produce stable electromagnetic torque, and have a wide range of speed regulation. During the starting and braking, the armature current of the motor generates electromagnetic torque, which can make full of the overload capacity of the motor and improve the starting and braking performance [20,21]. Therefore, although the vector control relies on the accuracy of the mathematical model, the drive motor using the control method has better system stability and faster response speed. Commonly used vector control methods include i_{d} = 0 control, cos ϕ = 1 control, maximum torque/current ratio control, field weakening control, and maximum efficiency control. Among them, the effect of i_{d} = 0 control is comparable to that of DC motors, especially in the lowspeed control range, the control effect is more prominent [22–24]. Therefore, the i_{d} = 0 control is the most used in the vector control method.
According to the development trend of the magnetic gear compound motor, a new dualmodulation magnetic gear compound motor (DMFMCM) applied to electric vehicles is proposed. The DMFMC is to improve the torque density of the magnetic gear compound motor, suppress the torque ripple, enhance the mechanical structure, and improve the reliability and stability of the motor during operation. The magnetic gear compound motor proposed in this paper combines the modulation pieces with the stator teeth to reduce an air gap, which is convenient for the assembly of the motor. Furthermore, it can realize the secondary speed modulation and increase the gear ratio. Then, the space vector pulse width modulation (SVPWM) method with i_{d} = 0 is used to control the speed of the DMFMCM so that the conversion of the motor speed is more stable. Based on the research of motor performance, this paper establishes mathematical model and finite elements analysis (FEA) cosimulation of DMFMCM to verify the accuracy of the mathematical model. In addition, a Simulink simulation model based on the mathematical model was established to verify the motor parameters during motor control. After ensuring the accuracy of the mathematical model, the SVPWM with i_{d} = 0 is adopted for DMFMCM, which makes the motor have better performance to meet the needs of the electric vehicles, and build the SVPWM control module simulation model.
2 Analysis and research of DMFMCM
The magnetic gear compound motor is a structure that can realize lowspeed and hightorque transmission, which has a good application prospect in the field of new energy such as wind power generation and electric vehicles. Figure 1 shows the topology of DMFMCM. The outer rotor of DMFMCM adopts a salient pole structure. When the convex pole coefficient δ_{e} = 0, it is a uniform permanent magnet; when the convex pole coefficient δ_{e} > 0, the permanent magnet is convex; when δ_{e} < 0, the permanent magnet is concave. The DMFMCM uses a convex pole factor of 0.5, i.e. the ratio of the protrusion of the permanent magnet to the width of the air gap is 0.5. The permanent magnets of the inner rotor have a tangential magnetized spoketype structure with pole shoes embedded between the permanent magnets, which can form a fluxmagnetic structure. In addition, the magnetic gear compound motor combines the modulated pieces and stator teethes, which makes the stator have the effect of magnetic field modulation. Therefore, the magnetogear compound motor reduces an air gap and achieves a twice modulating effect of the stator teeth and the modulation ring. Furthermore, this salient pole of the permanent magnets and the fluxmagnetic structure effectively enhance the torque density, increase the utilization of the permanent magnets and reduce the magnetic leakage. Therefore, the DMFMCM can effectively increase torque and suppress torque ripple [25,26].
The parameters of DMFMCM are shown in Table 1.
Fig. 1 Topology of DMFMCM. 
Parameters of DMFMCM.
2.1 Topology of DMFMCM drive system
The driving system of DMFMCM studied in this paper is a threephase 18slot/28pole field modulation permanent magnet motor (FMPMM). The threephase alternating current is applied to the winding coil of DMFMCM, which makes the energized winding generate a rotating magnetic field to couple with the FMPMM motor rotor and generate torque. The torque generated by the rotor of FMPMM is coupled with the outer permanent magnet rotor through the outer modulation ring so that the outer rotor generates torque to work.
The topological structure of the drive system FMPMM is shown in Figure 2. The rotor of FMPMM adopts a tangentially magnetized spoke structure, and the pole shoes are embedded between the permanent magnets to increase the torque. The stator adopts a slotted structure, and the stator teeth and the modulation pieces are combined to form a permanent magnet motor with field modulation. Due to the combination of the modulated block with the stator teeth, the two air gaps that would have been required with the addition of the modulating ring become one air gap. Therefore, the motor reduces the air gap and suppresses the magnetic leakage.
Fig. 2 Topology of FMPMM. 
2.2 The principle of torque generated by DMFMCM
2.2.1 The working principle of the magnetic gear
With the introduction of the modulation ring, the air gap space in the magnetic gear section is divided into two layers of the air gap. Due to the magnetic permeability of the modulation pieces is much greater than the air permeability, the magnetic path in the radial direction of the air gap is uneven, resulting in a change in the magnetic flux density in the air gap. The magnetic field modulation of modulation ring can be expressed by the air gap permeability function
where j represents the harmonics number, ω_{p} represents the modulation ring speed, ϕ_{θp} and ϕ_{rp} respectively represent the initial radial and tangential angles of modulation ring, , , and represent Fourier coefficients.
The permanent magnets of the inner rotor are embedded in poleshoes and distributed along the circumference. Therefore, the poleshoes and the modulation ring have a similar modulation effect. The specific expression is as follows
where k represents the harmonics number, n_{s} represents the poleshoes number, ω_{s} is the poleshoes speed, ϕ_{θs} and ϕ_{rs} respectively represent the initial radial and tangential angles of the poleshoes, , , and represent Fourier coefficients.
According to the modulating effect of the modulating commutation and pole shoe, The radial flux density distribution produced by the permanent magnet rotor is expressed as follows.
where b_{r}^{j,k,a} represents Fourier coefficients.
The tangential magnetic flux distribution produced by any rotor is calculated as follows.
where b_{θ}^{j,k,a} represents Fourier coefficients.
The spatial harmonic pole logarithm contained in the air gap magnetic field of the magnetic gear proposed in this paper is expressed as follows.
According to the operation principle of CMG in [8], the modulated polepieces number and PM polepairs should be satisfied as (8),
From the above equation, the poleshoes and outer rotor will generate p_{i} + n_{s} polepairs harmonics. Similarly, the modulated ring and outer rotor will produce p_{o} + n_{p} polepairs harmonics
The Fourier expansion of the magnetic induction intensity at a certain position is
where B_{r} represents the radial magnetic field component, B_{θ} represents the tangential magnetic field component. B_{rk} and B_{θk} respectively represent the kth Fourier coefficients of the radial and tangential component. α_{rk} and α_{θk} are the kth phase angles at that position. The harmonic torque of the k_{th} order is obtained according to Maxwell tensor method as
where r represents the airgap radius, µ_{0} represents the vacuum permeability, and l_{ef} represents the axis length.
2.2.2 The working principle of FMPMM
The electromechanical energy conversion of the magnetic gear compound motor is carried out by the magnetic field generated by the rotor and the armature winding. Based on the equivalent magnetic circuit method, the magnetic circuit can be analogous to an electric circuit to simplify the calculation and analysis process of the motor. It can be regarded as a loop composed of permanent magnets, magnetomotive force source and air gap. In order to simplify the electromagnetic torque derivation process of FMPMM, it is assumed that the magnetic field changes only in the crosssection, and the local magnetic saturation and flux leakage of the iron core are ignored. In the process of analyzing the change of air gap permeability, the equivalent air gap length δ_{g} ^{′} is as follows.
where δ_{g} is the length of the air gap, k_{c} is the Carter coefficient.
Due to the permanent magnet rotor using highperformance NdFeB, the demagnetization curve changes linearly, and the characteristics of the second quadrant can be approximated as a straight line [27]. Therefore, the permanent magnet can be equivalent to a magnetomotive force source with constant outer reluctance. The magnetic circuit analysis model of FMPMM and various parameters are shown in Figure 3a. Where θ_{m} is the thickness of the permanent magnet, r_{g} is the air gap radius, l_{m} is the radial length of the permanent magnet, and θ_{p} is the thickness of the rotor core. According to the known parameters, the permanent magnet magnetomotive force F_{m} can be calculated as follows.
where B_{r} is the remanence, μ_{0} is the vacuum permeability, and μ_{rm} is the relative permeability. The outer magnetic resistance R_{m} of the permanent magnet is calculated as (3).
where L_{a} is the axial length of FMPMM, and the equivalent air gap magnetoresistance R_{g} can be expressed as (14).
According to the magnetic flux low, it can be found that the magnetic flux of the rotor is calculated as follows.
The amplitude of air gap magnetomotive force can be expressed as follows.
Based on the calculation and analysis, the air gap magnetomotive force waveform of FMPMM can be equivalent to a square wave is shown in Figure 4. Considering the rotation of the rotor, and according to the Fourier transform, the calculation formula for the change of the air gap magnetomotive force F_{ag} with the circumferential position θ can be obtained as follows.
where p_{r} is the pole pairs number of the rotor, and the jth component amplitude F_{agj} is calculated as (18).
The above is a simplified analysis of the change in permeability caused by the change in the stator tooth slot with the aid of an equivalent air gap. In the following, the effect of the stator tooth slot change on the air gap permeability is considered, and the air gap permeability is calculated. Figure 5 shows the simplified air gap permeability waveform considering the stator tooth slot alternation. The permeability λ_{t} and λ_{s} at the corresponding air gap of the stator tooth slot can be expressed as follows.
where θ_{t} is the width of the stator teeth, and θ_{s} represents the width of the slots.
According to the permeance waveform and using Fourier transform, it can be seen that the calculation for the change of the air gap permeance with the position angle θ is as follows.
where λ_{0} is the DC component, N_{st} is the number of stator teeth, and the expression is as (22).
The harmonic component λ_{i} is calculated as follows.
In summary, it can be found that the air gap flux density of the DMFMCM is calculated as follows
where B_{h} is the highorder magnetic density harmonic component, which occupies a small proportion and it can be ignored. The formula shows that the fundamental component with the same number of rotor pole pairs, there are also N_{st−pr} and N_{st+pr} pairs of harmonic magnetic fields. In a threephase permanent magnet motor, the formula of the phase flux linkage is as follows.
where k_{d} is the fundamental distribution coefficient of the winding, N_{p} is the number of turns in series, and σ is the coil span. Furthermore, the stator slot pitch θ_{τ} can be expressed as (26).
The expression of the phase flux linkage of the DMFMCM can be obtained as shown in (27) by combining (24) and (25).
where q is the number of stator slots per pole.
The induced electromotive force of FMPMM is as follows.
The FMPMM is an 18slot/28pole permanent magnet synchronous machine. The winding pole pair p_{s} is 4, the gear ratio G_{r} between the inner rotor and the winding is 3.5, the number of stator slots per pole is 2.25, and the coil span σ is 2. Substituting the parameters into equation (18) above gives the phase induction potential amplitude of the FMPMM as follows.
In order to analyze the electromagnetic torque of FMPMM, using the brushless AC control method with i_{d} = 0, the electromagnetic torque T_{e} of the permanent magnet motor is expressed as follows.
In order to verify the accuracy of the above theory, an FEA analysis was performed on the FMPMM. The 2D model of FMPMM can be constructed using AutoCAD software and then imported into Maxwell FEA simulation software, or it can be created directly in Maxwell software. By feeding threephase currents into the windings and simulating the model, the FMPMM airgap flux density distribution and harmonic analysis are obtained in Figure 6. It can be found from the figure that the stator teeth will cause the airgap flux density to generate some spatial harmonics. In addition to the fundamental components with the same number of rotor pole pairs, N_{st−pr} and N_{st+pr} pairs harmonic component with larger amplitudes are generated. Therefore, it can be seen that the FEA analysis is consistent with the theoretically derived flux density expression (24). According to the equivalent circuit method of FMPMM and FEA, the working principle of the torque generated by the drive system of DMFMCM is determined to drive the entire compound motor to operate.
Fig. 3 (a) Magnetic circuit analysis model of FMPMM; (b) equivalent magnetic circuit of FMPMM. 
Fig. 4 Equivalent air gap magnetomotive force waveform of FMPMM. 
Fig. 5 The simplified air gap permeance waveform. 
Fig. 6 (a) Flux density waveform of FMPMM, (b) flux density harmonic of FMPMM. 
2.3 Comparison and analysis of DMFMCM and the conventional magnetic gear compound motor (CMGCM)
DMFMCM generates a rotating magnetic field by passing an alternating current, which couples with the permanent magnets on the spoke rotor through the stator teeth and drives the rotor to rotate. Then the rotating magnetic field is coupled with the outer rotor through the modulation ring to make the outer rotor rotate and work. In order to verify the superiority of DMFMCM, an FEA simulation model of the traditional 18slot magnetic gear compound motor (CMGCM) was established for comparison and analysis with DMFMCM. The FEA simulation results of the outer rotor torque of the two structures are compared to analyze their performance differences.
Based on the principle of quantitative comparison, the number of permanent magnets used to the two motors is the same, and the size of the motor are the same. The FEA is used to obtain the magnetic field distribution of two compound motors as shown in Figure 7.
As shown in Figure 8, the torque of DMFMCM is significantly higher than that of CMGCM, and the torque ripple of DMFMCM is significantly lower than that of CMGCM. According to (31), the torque ripple can be obtained in Table 2. It can be found that the static torque of DMFMCM is 321.02Nm, and it is 1.4 times that of CMGCM. Furthermore, the torque ripple of DMFMCM is 0.19%, which is 29.3% of CMGCM.
where T_{ripple} is the torque ripple, T_{max} is the maximum torque, T_{min} is the minimum torque, and T_{avg} is the average electromagnetic torque.
Fig. 7 (a) Magnetic field distribution of DMFMCM, (b) magnetic field distribution of CMGCM. 
Fig. 8 The torque of the outer rotor calculated by the FEA. 
Average electromagnetic torque and torque ripple.
3 Mathematical model of DMFMCM
The mathematical model of the drive motor is the basis of the simulation modelling and the analysis and design of the control system. Therefore, according to the electromagnetic characteristics of DMFMCM, a mathematical model under the twophase rotating coordinate system is established. The FEA method is used to verify the results of the mathematical model to ensure its accuracy. In addition, a simulation model of DMFMCM is built by Simulink components to verify the motor parameters of the control system.
The mathematical model of the threephase static coordinate system (abc coordinate system) based on the stator armature windings of the motor is shown in Figure 9. There are threephase symmetrical windings A, B, and C on the stator, which differ by 120° in electrical angle. d axis is on the direction of the permanent magnet pole axis of the rotor, and q is advanced by 90° in the counterclockwise direction of the rotor. Take a reference axis, the electrical angle between the d and the reference axis is θ. Based on the analysis method of the mathematical model, the flux linkage equation, voltage equation and torque equation of DMFMCM are established in abc coordinate system and the twophase rotating coordinate system (dq coordinate system).
Fig. 9 The abc coordinate system and dq coordinate system. 
3.1 Mathematical model in abc coordinate system
The total flux linkage of each winding of DMFMCM includes permanent magnet flux linkage and armature reaction flux linkage. Therefore, the flux equation expression of DMFMCM is as follows.
where ψ_{a},ψ_{b} and ψ_{c} are the total flux linkages of the threephase windings; i_{a}, i_{b}, i_{c} are the phase currents of the threephase windings; L_{aa},L_{bb},L_{cc} are the selfinductances of the phase windings; M_{ab},M_{ac},M_{ba},M_{bc},M_{ca},M_{cb} are the mutual inductances of the phase windings; ψ_{s} is the amplitude of the phase permanent magnet flux of the winding turn chain; θ is the electrical angle of the rotor.
When the brushless AC control mode with i_{d} = 0 is adopted, the threephase current of the motor is expressed as follows.
where I_{m} is the amplitude of the phase current.
The current flow direction is selected as the positive direction of the current, and the positive direction ofthe winding fluxlinkage and winding current conforms to the righthanded spiral rule. The threephase voltage balance equation of DMFMCM is as follows.
The resistance value of each phase is equal, R_{a} = R_{b} = R_{c} = R_{0}.
The electromagnetic torque formula under the electromagnetic equation is as follows.
The general formula of the mechanical motion equation of the motor is as follows.
where Tm is the total load torque of the system, J is the moment of inertia, and ω is the air gap speed.
The load torque is the torque of the entire transmission system, which can be interpreted as the mechanical torque obtained after the torque added to the outer rotor passes through the transmission system. Therefore, equation (36) is the mechanical equation for calculating the electromagnetic torque, which is the dynamics of a structure.
According to the mathematical model of DMFMCM in the abc coordinate system, the mathematical model in the dq coordinate system can be established, which lays the foundation for the simulation model of DMFMCM and the research of control strategy. In addition, based on the established mathematical model, a simulation model of the DMFMCM motor is built using Simulink simulation software.
3.2 Flux linkage equation in dq coordinate system
The inductance matrix of DMFMCM in the dq coordinate system is as follows.
where
L_{d}, L_{dq}, L_{d0}, L_{qd}, L_{q}, L_{q0}, L_{0d}, L_{0q} and L_{0} are the inductance component in the dq coordinate system.
Using the FEA method to calculate inductance can be divided into two steps. Firstly, the phase flux linkage ψ_{m} of the armature winding without current should be calculated; Secondly, the direct current I is applied to one phase of the armature winding, and the total flux linkage ψ of the phase winding is obtained. The total flux linkage is mainly composed of permanent magnet flux linkage and armature flux linkage, which can be expressed as follows.
The phase winding inductance can be calculated as follows.
The calculated inductance is selfinductance when the measured phase and the energized phase are the same phases, and the calculated inductance is mutual inductance when the measured phase and the energized phase are not the same phases. According to the method, the selfinductance and mutual inductance waveforms of phase A are shown in Figure 10.
The phase winding inductance of DMFMCM will periodically change with the change of the rotor position. According to the phase inductance calculated by the FEA method shown in Figure 10, when the rotor daxis advances the Aphase winding axis θ electrical angle, the threephase winding inductance can be expressed as follows.
where L_{D} is the average selfinductance of the phase, L_{S} is the amplitude of the phase selfinductance fundamental component, and M_{D} is the average of mutual inductance.
The calculation of inductance component is as follows.
According to (41), it can be seen that there is a small cosine component in L_{d} and L_{q}. Because DMFMCM adopts a spoketype inner rotor and a surfacemount outer rotor, it has a “salient pole effect”. Furthermore, due to the nonuniform circumferential reluctance caused by the alternating stator slots and the salient pole rotor, a part of the flux linkages on the daxis and the qaxis turns to each other. Therefore, the mutual inductance L_{dq} between the d and q axes is not 0 but a sine value.
Figure 11 shows the phase selfinductance and mutual inductance of DMFMCM in the abc coordinate system obtained by FEA. The inductance components L_{d}, L_{q}, L_{0} and L_{dq} calculated by FEA can be obtained according to (37). And the inductance component L_{d} ^{′}, L_{q} ^{′}, L_{0} ^{′} and L_{dq} ^{′} in the dq coordinate system derived by the mathematical model can also be calculated according to (37)–(39). The inductance waveforms calculated by FEA and the mathematical model are shown in Figure 12, and the average inductance is shown in Table 3. As shown in this table, the FEA shows that the average values of L_{d} and L_{q} are approximately equal, but the waveforms are different. It can be found that the DMFMCM has a salient pole effect. A comparison of the inductance waveforms in the dq coordinate system shows that the two waveforms are similar and have small errors, indicating that the inductance results derived from the mathematical model are accurate and valid.
According to the calculation of FEA and mathematical model, the results of the motor inductance in the dq coordinate system can be obtained as shown in the following Table 3.
It can be seen from (41) that the inductance components L_{d} and L_{q} of DMFMCM on the dq axis are not constant values, and the mutual inductances L_{dq} and L_{qd} are not zero. Therefore, the flux linkage in the dq coordinate system can be obtained as follows.
Substituting (40) into (42), the flux linkage in the dq coordinate system can be calculated as follows.
where ψ_{s} is the fundamental wave amplitude of the phase permanent magnet chain under noload in the abc coordinate system, where ψ_{md}, ψ_{mq}, ψ_{0} is calculated as follows.
Figure 13 shows the noload permanent magnet flux waveform of DMFMCM. The values of ψ_{md}, ψ_{mq}, and ψ_{m0} can be calculated, and the flux linkages in the dq coordinate system can also be calculated by substituting into (43) and (44).
In order to verify the accuracy of the flux linkages of the DMFMCM motor in the dq coordinate system, the paper adopts the id = 0 control method. The flux linkage waveforms on the daxis and qaxis are calculated by FEA and mathematical model, as shown in Figure 14. The figure shows that the waveforms calculated by FEA ψ_{d} and ψ_{q} are similar to the waveforms obtained by mathematical model ψ_{d} ^{′} and ψ_{q} ^{′}, and the average flux linkage is almost equal. Therefore, it can be found that the flux linkage of DMFMCM calculated by the mathematical model can be verified, and the accuracy of the mathematical model can be ensured.
Table 4 shows the average flux linkage of DMFMCM in the dq coordinate system calculated by the two methods. It can be found that the average flux linkages obtained by the two methods are almost equal to determining the daxis and qaxis flux linkage components. And through cosimulation, the calculation results of the mathematical model can be verified to ensure the accuracy of the mathematical model.
Fig. 10 Phase A inductance calculated by FEA. 
Fig. 11 (a) phase selfinductance and (b) phase mutual inductance in abc coordinate system. 
Fig. 12 (a) Selfinductance on dq axis, (b) mutual inductance on dq axis. 
Average inductance in coordinate system.
Fig. 13 The noload flux linkage of DMFMCM. 
Fig. 14 The flux linkage of DMFMCM on dq axis. 
Average flux linkage in dq coordinate system.
3.3 Voltage equation in coordinate system
Due to the torque control adopting alternating current, the current in the dq coordinate system obtained by Park transformation in the abc coordinate system is as follows.
where α is the current factor angle, I_{m} is the current amplitude, and when α = 0, it is i_{d} = 0 AC control.
Combining the flux equation and the current equation, the voltage equation of DMFMCM can be calculated.
where ω is the rotational angular velocity of the airgap magnetic field.
3.4 Torque equation in dq coordinate system
According to the voltage and current equations in the dq coordinate system, the electromagnetic torque in the dq coordinate system can be calculated as follows.
The torque waveform of the DMFMCM obtained after using the mathematical model with i_{d} = 0 and finite element analysis described in this paper is shown in Figure 15. By analyzing and calculating the torque using FEA and mathematical model cosimulation, the torque and torque ripple of DMFMCM can be determined more accurately. As shown in Table 5, the torque waveforms calculated by the two methods are similar, and the torque ripple is also low.
According to the torque obtained in the figure, combined with the (31), the torque ripple calculated by the two methods can be calculated. The calculation results are shown in Table 5.
Based on the mathematical model, the stator winding of the DMFMCM is modelled using Simulink. According to the needs of SVPWM, the whole model adopts a modular design. The stator winding of the DMFMCM requires output current, electromagnetic torque, and rotational angular velocity. Each module is independent and communicated through shared variables. The relationship of the modules can be intuitively seen from the Simulink simulation model of stator windings for DMFMCM and determine the relationship between the parameters. The specific simulation model is shown in Figure 16.
According to the mathematical model, Simulink is used to model the DMFMCM motor. As shown in Figure 16b, threephase alternating current and total load torque T_{m} are applied to the motor. After a series of motor coupling actions, threephase AC I_{abc}, electromagnetic torque T_{e} and mechanical angular velocity ω_{m} can be obtained.
Fig. 15 The torque calculated by FEA and mathematical model. 
Electromagnetic torque and torque ripple.
Fig. 16 (a) The stator winding model of the DMFMCM simulated by Simulink and (b) the module package of DMFMCM. 
Fig. 17 The relationship between reference vector and space vector. 
4 SVPWM control system
4.1 The principle of SVPWM control system
Space vector pulse width modulation (SVPWM) is a new control strategy for controlling the converter based on the space voltage vector switching of the converter. The core idea of the SVPWM algorithm is to use the inverter space voltage vector switching to obtain a standard circular rotating magnetic field, which enables the AC motor to better control performance by the condition of a low switching frequency. The theoretical basis is the principle of average value equivalence, which combines the voltage vectors within a switching period T_{s} to make the average value equal to the given voltage vector. According to the working principle of SVPWM, the specific implementation process of the SVPWM algorithm includes three steps: (1) judgment of voltage vector, (2) determination of voltage working time and (3) determination of sector vector switching point.
4.1.1 Judgment of voltage vector
In order to simplify the judgment process, the voltage is equivalently judged. Where U_{β} = U_{ref} sin θ, U_{α} = U_{ ref} cos θ Figure 17.
Define intermediate variables A, B, and C, with the following judgments.
If U_{a} > 0, then A = 1, otherwise A = 0.
If U_{b} > 0, then B = 1, otherwise B = 0.
If U_{c} > 0, then C = 1, otherwise C = 0.
The sector calculation formula is as follows.
The sector of the reference voltage vector is determined by N, and the corresponding relationship is shown in Table 6.
The relationship between N and sector.
4.1.2 Determination of voltage working time
According to the sector where the obtained voltage vector is located, the action time of the space vector can be calculated.
The U_{dc} is the DC bus voltage, and the assignment relationship corresponding of T_{1}, T_{2}, and T_{3} is obtained as shown in Table 7.
Saturation judgment is performed on the copied T_{1} and T_{2}. If T_{1}+T_{2} < T, then T_{1} and T_{2} keep unchanged; if T_{1}+T_{2} > T, the relationship is as follows.
The relationship between the action time and N.
4.1.3 Determination of sector vector switching point
According to SVPWM algorithm, it is defined as follows.
The relationship between the sector vector switching points T_{cm1}, T_{cm2}, T_{cm3} and N can be determined as shown in Table 8.
Time switching point of each sector.
4.2 Modeling and simulation of SVPWM
According to the results obtained by the SVPWM algorithm, the triangular carrier signal is compared with the vector switching points of each sector to output a PWM pulse signal. The SVPWM is mainly to control the rotor magnetic field. In the rotating dq coordinate system, the magnetic current and torque current are independently controlled to improve the control performance of DMFMCM. In addition, the mathematical model of DMFMCM in the dq coordinate system is established. When i_{d} = 0, the electromagnetic torque of DMFMCM is only related to the variable i_{q}, thereby improving the speed regulation performance of the system. The functional block diagram of SVPWM is shown in Figure 18.
As shown in Figure 18, the control strategy includes the current loop, speed loop and SVPWM algorithm, and the SVPWM control of DMFMCM is mainly the control of the current component on the dq coordinate system. The difference value between the reference speed and the actual speed after poweron is adjusted by the PI regulator, and output is the given value of the qaxis current i_{q} ^{′}. The threephase currents i_{a}, i_{b} and i_{c} from DMFMCM are transformed by Clark and Park to obtain the current components i_{d} and i_{q} in the dq coordinate system. In addition, the given value i_{q} ^{′} is compared with the actual value i_{q}, and then the qaxis voltage U_{q} ^{′} adjusted by the PI regulator can be output. In the same way, the daxis goes through the same process, and the daxis voltage U_{d} ^{′} is output. U_{q} ^{′} and U_{d} ^{′} are transformed by Park to obtain U_{α} and U_{β}, and then the PWM signal is obtained by the SVPWM algorithm to control the inverter. The simulation model of SVPWM established by Simulink is shown in the Figure 19.
The conditions of the simulation model are set as follows: reference speed n ^{′} = 700 r/min, PWM switching frequency f = 50 Hz, simulation time is 0.75 s, variable step size is ode23tb algorithm, relative error is 1e–4. The simulation parameters of DMFMCM are shown in Table 9, the results are shown in Figure 20.
According to the simulation results shown in Figure 18, the threephase DMFMCM starts without load. The speed adjusted by the PI regulator will rapidly increase from 0 and exceed the reference speed within 0–0.05 s and will quickly stabilize at the reference speed. At the same time, the torque suddenly increased and stabilized at 16.13 Nm within 0.05 s. At 0.2 s, the rotational speed changes abruptly and immediately stabilize at the reference speed. And the torque also jumped to 25.34 Nm and stabilized. In addition, changes in current are also synchronized with changes in speed and torque. It can be seen that when the electric vehicle starts and shifts, it can output stable torque in a short time, and when shifting and adjusting the speed, the output torque is quickly stable. Therefore, the simulation results show that the SVPWM control is based on PI regulation, which makes DMFMCM have the advantages of startup and shifting time short, high stability, strong antiinterference ability, and good speed regulation performance.
Fig. 18 The functional block diagram of the SVPWM system with i_{d} = 0. 
Fig. 19 The SVPWM model of DMFMCM based on i_{d} = 0. 
Simulation parameters of DMFMCM.
Fig. 20 (a) the simulation curve of motor speed r, (b) the simulation curve of threephase stator current I_{abc}, and (c) the simulation curve of motor torque T_{e}. 
5 Conclusion
The paper analyzes and studies a new DMFMCM used in electric vehicles. Compared with the conventional magnetic gear compound motor (CMGCM), the DMFMCM has the advantages of strong mechanical strength, high torque density, and low torque ripple. The torque of the outer rotor is 321.02Nm, and the output torque of the outer rotor is 1.4 times that of CMGCM. Furthermore, the torque ripple of the outer rotor is 0.19%, which is only 29.3% of CMGCM. Therefore, the electromagnetic performance of DMFMCM is much better than CMGCM.
The mathematical model of DMFMCM is established, and it is also modelled by Simulink software. The flux linkages equation, electromagnetic torque and voltage equation obtained based on the mathematical model are compared with the FEA simulation results of DMFMCM. According to the FEA and mathematical model cosimulation, the accuracy of the mathematical model is verified through the FEA and mathematical model cosimulation, which lays the foundation for SVPWM control.
This paper establishes a Simulink simulation model of DMFMCM and outputs the results of speed, current and electromagnetic torque based on the input parameters of DMFMCM. It provides a basis for the SVPWM control and verifies the parameters of DMFMCM. By analyzing the principle of SVPWM control, the control strategy mainly includes the speed control loop, current control loop, and SVPWM algorithm. In addition, the system adopts the AC control mode with id = 0, builds a threephase DMFMCM motor control system model based on PI regulation, and outputs threephase current, electromagnetic torque and speed. The simulation results show that the SVPWM control strategy makes DMFMCM have the advantages of short startup shift time, high stability, strong antiinterference ability, and good speed regulation performance, which meets the development needs of electric vehicle drive motors.
Conflict of Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Financial support
This work was supported by the National Natural Science Foundation of China (Grant No. 51765020), and the Natural Science Foundation of Jiangxi Province (Grant No. 20161BAB206153).
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Cite this article as: J.G. Wang, B. Zhang, L.Q. Qian, Research and control of a new dualmodulation magnetic gear compound motor for electric vehicles based on a mathematical model and FEA cosimulation, Mechanics & Industry 24, 18 (2023)
All Tables
All Figures
Fig. 1 Topology of DMFMCM. 

In the text 
Fig. 2 Topology of FMPMM. 

In the text 
Fig. 3 (a) Magnetic circuit analysis model of FMPMM; (b) equivalent magnetic circuit of FMPMM. 

In the text 
Fig. 4 Equivalent air gap magnetomotive force waveform of FMPMM. 

In the text 
Fig. 5 The simplified air gap permeance waveform. 

In the text 
Fig. 6 (a) Flux density waveform of FMPMM, (b) flux density harmonic of FMPMM. 

In the text 
Fig. 7 (a) Magnetic field distribution of DMFMCM, (b) magnetic field distribution of CMGCM. 

In the text 
Fig. 8 The torque of the outer rotor calculated by the FEA. 

In the text 
Fig. 9 The abc coordinate system and dq coordinate system. 

In the text 
Fig. 10 Phase A inductance calculated by FEA. 

In the text 
Fig. 11 (a) phase selfinductance and (b) phase mutual inductance in abc coordinate system. 

In the text 
Fig. 12 (a) Selfinductance on dq axis, (b) mutual inductance on dq axis. 

In the text 
Fig. 13 The noload flux linkage of DMFMCM. 

In the text 
Fig. 14 The flux linkage of DMFMCM on dq axis. 

In the text 
Fig. 15 The torque calculated by FEA and mathematical model. 

In the text 
Fig. 16 (a) The stator winding model of the DMFMCM simulated by Simulink and (b) the module package of DMFMCM. 

In the text 
Fig. 17 The relationship between reference vector and space vector. 

In the text 
Fig. 18 The functional block diagram of the SVPWM system with i_{d} = 0. 

In the text 
Fig. 19 The SVPWM model of DMFMCM based on i_{d} = 0. 

In the text 
Fig. 20 (a) the simulation curve of motor speed r, (b) the simulation curve of threephase stator current I_{abc}, and (c) the simulation curve of motor torque T_{e}. 

In the text 
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