The outer rotor of a permanent magnet assisted dual rotor motor is a surface-mounted permanent rotor, which shares several characteristics with permanent magnet dual rotor motors. The distinction is that the permanent magnet synchronous motor and the synchronous reluctance motor are integrated into a single motor, and the synchronous reluctance rotor is used to replace the internal permanent magnet rotor (Yunyun et al., 2012). The two rotors share a shaft and rotate at the same mechanical speed. The stator winding adopts the ring winding method, and the magnetomotive force direction of the stator winding of the outer motor is opposite that of the stator winding of the inner motor. In order to achieve magnetic insulation between the inner stator and the outer stator, a magnetic insulator used as a barrier ring is used in the middle of the stator yoke. The insulator is made of brass. Figure 1 is a three-dimensional diagram of the motor. The parameters of the motor are shown in Table 1.

In order to ensure that the motor outputs ideal torque, the internal synchronous reluctance rotor should be properly designed. The typical number of rotor poles of a synchronous reluctance motor is four, but this requires the external surface-mounted permanent magnet rotor to also maintain a four-pole topology. This design method will cause the external permanent magnet rotor yoke to become thicker, and with added weight comes reduced performance. The external surface-mounted permanent magnet rotor is more suitable for the application of multi-rotor pole topology to reduce the thickness of the yoke and the weight of the motor, but this type of topology is not the best design for synchronous reluctance motors because, because of the multi-pole structure, the pole speed of the magnetization inductor of the synchronous reluctance rotor will decrease, and the performance of the synchronous reluctance rotor will also decrease under the condition of a high pole number. Therefore, the number of rotor poles selected for this type of motor is six, and the number of poles of the inner and outer rotors is the same, so as to balance the low number of magnetic poles of the synchronous reluctance rotor and the high number of magnetic poles of the surface-mounted permanent magnet rotor.

The design of the synchronous reluctance rotor flux barrier focuses on the number and position of each flux barrier layer. Generally, for the inner rotor synchronous reluctance motor, the torque ripple decreases with the increase of the number of flux barriers, but due to the limitations of the manufacturing process, the rotor flux barriers are generally set to 4 layers in the design of the synchronous reluctance motor (Jurca and Martis, 2017), so the number of rotor magnetic barriers in this paper is also set to 4 layers.

Different windings, cogging slots, permanent magnet arrangements, and reluctance rotor magnetic barrier shapes of permanent magnet-assisted dual-rotor motors have various topological structures. The permanent magnet rotor can not only be installed on the surface but also be buried in the ground, and the shape of the magnetic barrier of the reluctance rotor can also be LAL, ARC, segmented rectangle, etc. Figure 2 shows the permanent magnet rotor and a few examples of magnetic choke rotors.

A major contributor to torque ripple in permanent magnet motors is the cogging torque (Wang et al., 2011). In order to reduce the influence of the cogging torque on the surface-mount permanent magnet motor, Patel (2023) recommended slotting to reduce the gear toque by 48.78%. Rashid and Mohammed (2023) proposed a sawtooth moment mitigation method based on the radial slits of the pole pieces. Won et al. (2023) used neural network prediction to achieve the purpose of reducing cogging torque. Wang et al. (2023) adopted the method of combining in-phase unit (IPU) grouping and slotting angle offset so that the main harmonic component of the sawtooth torque is cancelled to achieve the purpose of reducing the cogging torque.

This paper chooses to determine the factors affecting cogging torque from the definition of cogging torque. Cogging torque T c o g is defined as the negative derivative of the magnetic energy w with respect to the position angle α when de-energized. The expression can be derived as follows: T c o g α = π z L e 4 u 0 R 2 2 − R 1 2 ∑ n = 1 ∞ n G n B r n sin n z α (11)

In the formula, R 1 is outside armature diameter of external motor, R 2 is stator inner diameter of external motor, z is number of stator slots, α is the relative position angle between the stator and rotor, u 0 is vacuum permeability.

B r n is the Fourier coefficient of the residual flux density squared along the circumference of permanent magnets, and the formula is as follows: B r n = 2 n π B r 2 ⁡ sin ⁡ n α p π (12)

In the formula, B r is the residual flux density in a permanent magnet, α p is polar arc coefficient of permanent magnetic pole.

G n is the Fourier coefficient of the square relative permeability of the air-gap and the formula is as follows: G n = ∑ n = 1 ∞ 2 n π h m h m + δ 2 ⁡ sin n π − n z θ s 2 (13)

In the formula, δ is air gap length, h m is the thickness of the permanent magnet along the magnetization direction, θ s is the corresponding radian value of the width of the stator groove.

It can be seen from the above formula that the slot opening angle width, magnetic steel angle width, magnetic steel thickness, and air gap length all have a great influence on the cogging torque. Therefore, these four parameters are selected as the cogging torque optimization parameters in this paper to study their effect on cogging torque. Table 2 lists the optimized variable parameter ranges.

In order to explore the law of cogging torque weakening caused by different design variables, each design variable is parameterized in the optimization process, and one parameter is selected as a variable. The upper and lower limits are given as shown in Table 2. Other parameters are constants; after obtaining the waveforms of the cogging torque in various situations, select the maximum value to draw the trend of the cogging torque changing with the parameters, and select the optimal parameters for simulation verification in Maxwell. If the optimal effect meets expectations, the optimization is complete. If the effect is not as expected, it is necessary to change the upper and lower limits of the parameters or to optimize the parameters. The specific optimization scheme flow chart is shown in Figure 3. As we can see in Figure 4, the optimal values of the four parameters can be easily found according to the graph.

During the cogging torque parameterization, different degrees of attenuation can be demonstrated by using different methods and changing the parameter range. It can be seen from Figure 4 that when the pole arc coefficient is between 0.72 and 0.74 and the thickness of the magnetic steel is between 1.6 mm and 1.8 mm, the calculated cogging torque is relatively lower. Considering that increasing the air gap will reduce the output torque of the motor, the length of the air gap should be kept between 0.6 mm and 1 mm. The cogging torque waveform before and after adjustment is shown in Figure 5. The cogging torque is reduced from 1.5 Nm to 0.15 Nm, and the optimized output torque waveform is more stable, the torque ripple is lower, and the optimization effect is more obvious.

After the optimization is completed, use the finite element software Maxwell to establish a two-dimensional model of the motor, set the rated current on the inner and outer stator windings, and obtain the magnetic flux density cloud diagram of the permanent magnet assisted dual-rotor motor with load through finite element simulation. As shown in Figure 6, the maximum value of the magnetic flux density appears at the position of the inner and outer stator yoke, which is about 1.8T, which does not exceed the design limit of 2T and meets the motor operation specification.

The air-gap magnetic field density is an important indicator of motor design. Figure 7 is the air-gap flux density curve of the internal and external motors under no-load and rated current. The amplitude of the air-gap flux density reflects the magnetic field strength and torque density of the motor, it can be seen that the waveform is closer to a sine wave. From the comparison of the two curves, we can see that the air gap flux density curve of the inner motor is more reasonable than the sinusoidal distribution of the outer motor, indicating that the inner motor can meet the requirements of torque output and power density. The peak values of the air gap magnetic density of the outer motor and the inner motor are 0.8T and 1 T, respectively, which generally meet the design requirements.

Since the motor has a double-air-gap structure, its total output torque is the sum of the output torques of the inner and outer motors, and the torque output waveforms of the inner and outer motors are shown in Figure 8. According to the output torque results, the output torque of the inner rotor synchronous reluctance motor is about 4  Nm , and the torque ripple is about 28.8%. The output torque of the outer rotor permanent magnet synchronous motor is about 13  Nm , and the torque ripple is about 17.1%, which verifies the effectiveness of the optimized design, and the quality of the motor output torque has been significantly improved.

The dual-rotor motor is composed of inner rotor, outer rotor, permanent magnet, magnetic isolation ring and other parts. It has the characteristics of strong coupling and nonlinearity. For the convenience of analysis, the assumptions are as follows: 1) Neglect eddy current and hysteresis loss; 2) Excitation current There is no response time; 3) The stator windings are star-connected, and the stator windings are connected in parallel. According to the above assumptions, it can be obtained that the dual-rotor motor can be split into a permanent magnet synchronous motor and a synchronous reluctance motor, and after transformation by clark and park the voltage equation of the motor under the dq axis. u d 1 u q 1 = cos θ 1 cos θ 1 − 2 π 3 cos θ 1 − 4 π 3 − sin θ 1 − sin θ 1 − 2 π 3 − sin θ 1 − 4 π 3 U A 1 U B 1 U C 1 (14) u d 2 u q 2 = cos θ 2 cos θ 2 − 2 π 3 cos θ 2 − 4 π 3 − sin θ 2 − sin θ 2 − 2 π 3 − sin θ 2 − 4 π 3 U A 2 U B 2 U C 2 (15)

Since the double-rotor motor is divided into two parts, the inner rotor and the outer rotor, the electromagnetic torque of the motor is expressed as the permanent magnet torque of the outer motor and the reluctance torque of the inner motor, and the sum of the inner and outer torques is the total electromagnetic torque of the motor. T e = T e 1 + T e 2 = 3 2 p 1 L d 1 − L q 1 i d 1 i q 1 + 3 2 p 2 i q 2 ψ f (16) T e − T L = J d ω d t + B ω (17)

In these formulas, p x is the logarithm of the motor poles, U A x U B x U C x is the motor stator three-phase current, i d x i q x is current of d q axial, u d x u q x is voltage of d q axial, θ x is angle from a b c axial to d q axial, T e x is the electromagnetic torque of a motor, T L is a load torque, B is the damping coefficient, ω is the mechanical speed of a motor, J is the moment of inertia of the motor, ψ f is the permanent magnet linkage of a motor. x equals to 1 is the reluctance part of the motor, x equals to 2 is the permanent magnet portion of the motor.

Vector control is based on the control idea of a DC motor, converts the three-phase current into excitation current and torque current, and realizes indirect control of the motor by changing the magnitude of the excitation current and torque current, that is, by controlling the axis current of the motor to achieve the purpose of controlling the motor. At present, the commonly used vector control methods of permanent magnet synchronous motors include i d = 0 control, cos ⁡ φ = 1 control, maximum torque-current ratio control and flux weakening control; vector control methods of synchronous reluctance motor include maximum torque-to-current ratio control, maximum power factor control, maximum torque change rate control, and so on. The external motor of the double-rotor motor is a surface-mounted permanent magnet synchronous motor, and the i d = 0 control is equivalent to the maximum torque-to-current ratio control, therefore, the inner and outer motors of the dual-rotor motor in this paper are controlled by the maximum torque-current ratio, which simplifies the control system.

Due to the existence of the magnetic isolation ring, the dual-rotor motor can be equivalent to a surface-mounted permanent magnet synchronous motor and a synchronous reluctance motor. Conventional control requires two sets of controllers that interact with each other. In order to solve this problem, torque distribution is introduced. Torque distribution is used to fix the ratio of the output torque of the internal and external motors, distribute the torque current of the internal and external motors, and simplify the control system.