A mathematical modulation model and magnetic field analysis method for dual permanent magnet vernier motor

Abstract

The present invention discloses a mathematical modulation model and magnetic field analysis method for a dual permanent magnet vernier motor. First, a mathematical model of the stator and rotor modulation operators is established based on the dual permanent magnet vernier motor object to be analyzed; then, based on the position and size parameters of the permanent magnets and armature windings, equivalent square wave mathematical models of the initial stator permanent magnet magnetomotive force, the initial rotor permanent magnet magnetomotive force, and the initial armature magnetomotive force are respectively established; secondly, based on the modulation relationship of each part, each initial magnetomotive force is multiplied by one or more modulation operators that participate in the corresponding action to solve the actual magnetomotive force component of each part in the air gap; at this time, the air gap has been equivalently converted to a uniform air gap, and then, based on the actual magnetomotive force of the air gap, the magnetic field component of each part in the air gap is obtained to realize the magnetic field analysis of the motor; finally, the magnetic field analyzed by the mathematical modulation model is compared with the finite element software to verify the feasibility of the present invention. The present invention does not rely on simulation software, and the calculation and analysis are convenient, flexible, and fast.

Description

Technical Field

[0001] The invention relates to a mathematical modulation model of a dual permanent magnet vernier motor and a magnetic field analysis method, and belongs to the field of mathematical modeling and electromagnetic field analysis calculation of motors. Background Art

[0002] Dual permanent magnet vernier motors, with permanent magnets distributed on both the stator and rotor, offer higher torque density than conventional permanent magnet vernier motors of the same size. They are suitable for low-speed, high-torque applications such as wind power generation, oilfield extraction, and radar turntables. However, due to their unique bilateral excitation structure and bidirectional magnetic field modulation, dual permanent magnet vernier motors exhibit more complex modulation relationships and richer harmonic content than conventional permanent magnet vernier motors, making modeling and analysis difficult.

[0003] At present, the analysis methods for magnetic field modulation motors such as dual permanent magnet vernier motors mainly rely on finite element method (FEM) software developed abroad such as ANSYS Maxwell and JMAG. FEM software uses the principle of automatic meshing to decompose the entire problem area, so that each sub-area becomes a simple part and then solves it. It not only has high calculation accuracy, but also can adapt to complex shapes and is widely used in the field of motor design. However, FEM software takes a long time to solve, and it is difficult to reveal the operating mechanism of this type of motor, which is not conducive to in-depth study and analysis of the motor. The Chinese patent publication number CN 108595772A, entitled "A 2D magnetic circuit decomposition modeling method for stator and rotor dual permanent magnet vernier motors", discloses a dual permanent magnet vernier motor modeling and analysis method that uses an equivalent magnetic network for magnetic circuit decomposition. This method is simple to calculate and improves design efficiency through shorter time consumption. However, this method still requires the use of software such as MATLAB for iterative calculations. When the motor model changes, it needs to be remodeled and analyzed, which is rather troublesome and still cannot reveal the electromagnetic field action mechanism of this type of motor in principle. Summary of the Invention

[0004] The purpose of the present invention is to propose a mathematical modulation model and magnetic field analysis method for a dual permanent magnet vernier motor. The motor's mechanical structure and electromagnetic field model are characterized by mathematical formulas, without relying on simulation software for analysis. When the motor size parameters change, it can be flexibly adjusted, and the electromagnetic field action mechanism of this type of motor can be explained in principle.

[0005] To achieve the above-mentioned purpose, the technical solution adopted by the present invention is as follows: the selected dual permanent magnet vernier motor example is a stator with 12 slots / stator permanent magnets with 12 poles / rotor permanent magnets with 19 poles, including six parts: stator core, stator permanent magnets, rotor core, rotor permanent magnets, stator winding, and air gap; the stator core contains 12 split teeth, each split tooth is split into 2 small teeth, participating in modulation; the 12 continuous-pole permanent magnets of the stator are respectively embedded in the slots generated by each split tooth; the rotor core contains 19 salient poles, participating in modulation; the 19 continuous-pole permanent magnets of the rotor are respectively embedded in the slots generated by the salient poles; the stator armature winding adopts fractional slot concentrated winding; the air gap is located between the stator and the rotor, and the air gap thickness is 0.7mm.

[0006] Furthermore, according to the dimensional parameters of the motor stator and rotor structures, mathematical models of the stator permanent magnet slot modulation operator, the stator winding slot modulation operator and the rotor permanent magnet slot modulation operator are established respectively; the modulation operator satisfies the linear rate, the stator modulation operator is the linear multiplication of the stator winding slot modulation operator and the stator permanent magnet slot modulation operator, and the rotor modulation operator is the rotor permanent magnet slot modulation operator; the modulation operator is a mathematical function with a value between 0 and 1, which characterizes the drop of the magnetomotive force at each slot, so that the uneven air gap caused by the stator split teeth and the rotor salient poles is equivalent to a uniform air gap.

[0007] Furthermore, according to the parameters of the stator and rotor permanent magnets, the initial stator permanent magnet magnetomotive force and the initial rotor permanent magnet magnetomotive force models are established respectively. For the convenience of modeling, the initial magnetomotive force can be simplified and equivalent to a square wave.

[0008] Furthermore, according to the position distribution of the stator armature three-phase winding, the initial magnetomotive force of each phase is superimposed to establish an initial armature magnetomotive force model. For the convenience of modeling, the initial armature magnetomotive force can be simplified to be equivalent to a square wave and decomposed into a forward component and a reverse component.

[0009] Furthermore, considering the continuous-pole permanent magnet structure of the stator of the dual permanent magnet vernier motor, the modulation process of the stator permanent magnet magnetomotive force no longer needs to consider the modulation effect of the stator permanent magnet slots. The initial stator permanent magnet magnetomotive force is multiplied by the stator winding slot modulation operator to obtain the stator adjusted stator permanent magnet magnetomotive force, and the stator adjusted stator permanent magnet magnetomotive force is multiplied by the rotor modulation operator to obtain the actual stator permanent magnet magnetomotive force.

[0010] Furthermore, considering the continuous-pole permanent magnet structure of the dual permanent magnet vernier motor rotor, the modulation process of the rotor permanent magnet magnetomotive force no longer needs to consider the modulation effect of the rotor. The actual rotor permanent magnet magnetomotive force can be directly obtained by multiplying the initial rotor permanent magnet magnetomotive force by the stator modulation operator.

[0011] Furthermore, the modulation process of the armature magnetomotive force must take into account the modulation effects of the stator and rotor at the same time. The initial armature magnetomotive force multiplied by the stator modulation operator can obtain the stator-modulated armature magnetomotive force, and the stator-modulated armature magnetomotive force multiplied by the rotor modulation operator can obtain the actual armature magnetomotive force.

[0012] Furthermore, the actual magnetomotive force multiplied by the uniform air gap coefficient μ0 / g can be used to obtain the magnetic flux density. The stator permanent magnet magnetic field and the rotor permanent magnet magnetic field are superimposed to obtain the no-load air gap magnetic field. The no-load air gap magnetic field and the armature magnetic field are superimposed to obtain the loaded air gap magnetic field. Based on the analyzed magnetic field, electromagnetic parameters such as induced electromotive force and torque can be obtained.

[0013] The beneficial effects of the present invention are:

[0014] 1. The present invention adopts mathematical models to respectively perform equivalence on the stator and rotor cores, stator and rotor permanent magnets, windings and other structures of the dual permanent magnet vernier motor. The motor can be analyzed without the aid of any finite element or other analysis software, which is more convenient and quick.

[0015] 2. The present invention introduces a modulation operator in the modeling to consider the complex bilateral modulation effect of the dual permanent magnet vernier motor, thereby simplifying the modulation analysis process. At the same time, the modulation operator can take into account the drop and uneven distribution of the slot magnetomotive force, which is more accurate than the traditional square wave magnetic permeability model.

[0016] 3. The present invention equates the actual air gap magnetic field of the dual permanent magnet vernier motor to the superposition of the magnetic fields generated by the modulation of the three initial magnetic fields of the stator, rotor and armature. The analysis process is highly logical and easy to understand.

[0017] 4. The present invention does not require re-modeling and analysis like finite element or magnetic network methods after the motor slot-pole matching or size parameters change. It only needs to change the size of the corresponding parameters in the mathematical model, which is very flexible and convenient.

[0018] 5. The present invention reveals the electromagnetic field action mechanism of the dual permanent magnet vernier motor in principle, and can establish an accurate magnetic field analytical model at the same time. It has the advantages of clear and intuitive physical concepts and simple and clear mathematical analysis process, and can provide theoretical guidance for the design and structural innovation of this type of motor. BRIEF DESCRIPTION OF THE DRAWINGS

[0019] Figure 1 is a structural diagram of an example of a motor used in the present invention;

[0020] Figure 2 yes Figure 1 Diagram of the equivalent mathematical model of the stator modulation operator;

[0021] Figure 3 yes Figure 1 Diagram of the equivalent mathematical model of the rotor's modulation operator;

[0022] Figure 4 yes Figure 1 Diagram of the equivalent mathematical model of the initial magnetomotive force of the stator permanent magnet;

[0023] Figure 5 yes Figure 1 Diagram of the equivalent mathematical model of the initial magnetomotive force of the rotor permanent magnet;

[0024] Figure 6 yes Figure 1 Diagram of the equivalent mathematical model of the initial magnetomotive force of the armature winding;

[0025] Figure 7 This is a schematic diagram comparing the stator permanent magnet magnetic field obtained by finite element software and mathematical modulation model;

[0026] Figure 8 This is a schematic diagram comparing the rotor permanent magnet magnetic field obtained by finite element software and mathematical modulation model;

[0027] Figure 9 This is a schematic diagram comparing the armature magnetic field obtained by finite element software and the mathematical modulation model;

[0028] Figure 10 It is a process diagram of modeling and analysis of the present invention;

[0029] In the figure: 1. Stator core; 2. Rotor core; 3. Stator permanent magnet; 4. Rotor permanent magnet; 5. Armature winding; 6. Air gap. DETAILED DESCRIPTION

[0030] The specific implementation of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention.

[0031] Figure 1 The specific dual permanent magnet vernier motor structure diagram combined with the present invention, the selected dual permanent magnet vernier motor example is a stator with 12 slots / stator permanent magnets with 12 poles / rotor permanent magnets with 19 poles; wherein 1 is the stator core, 2 is the rotor core, 3 is the stator permanent magnet, 4 is the rotor permanent magnet, 5 is the armature winding, and 6 is the motor air gap; the stator core contains 12 split teeth, each split tooth is split into 2 small teeth, participating in modulation; the 12 continuous-pole permanent magnets of the stator are respectively embedded in the notches produced by each split tooth, and the magnetization direction of the permanent magnets is radially away from the center of the circle; the rotor core contains 19 salient poles, participating in modulation; the 19 continuous-pole permanent magnets of the rotor are respectively embedded in the notches produced by the salient poles, and the magnetization direction of the permanent magnets is radially away from the center of the circle; the stator armature winding adopts fractional slot concentrated winding; the air gap is located between the stator and the rotor, and the air gap thickness is 0.7mm.

[0032] Figure 10 The modeling and analysis process of the present invention is briefly summarized, including the following steps:

[0033] Step 1: Establish mathematical models of stator permanent magnet slot modulation operator, stator winding slot modulation operator and rotor permanent magnet slot modulation operator.

[0034] Mathematical models of the stator permanent magnet slot modulation operator, stator winding slot modulation operator and rotor permanent magnet slot modulation operator are established; the modulation operator satisfies the linear rate, the stator modulation operator is the linear multiplication of the stator winding slot modulation operator and the stator permanent magnet slot modulation operator, and the rotor modulation operator is the rotor permanent magnet slot modulation operator; the modulation operator is a mathematical function with a value between 0 and 1, which represents the drop of the magnetomotive force at each slot, so that the uneven air gap caused by the stator split teeth and the rotor salient poles is equivalent to a uniform air gap.

[0035] Figure 2 The equivalent mathematical model diagram of the stator modulation operator is shown in Figure 2. The stator permanent magnet slot modulation operator M sm (θ) can be expressed as:

[0036]

[0037] Where v is a positive integer, P s is the number of stator permanent magnet slots; a sm0 、a smv are the DC and AC components of the Fourier coefficients respectively, which can be calculated based on the stator permanent magnet slot parameter b m , t m , g and the intermediate variable α they constitute m , ε1 is obtained, which is expressed as:

[0038]

[0039]

[0040] Stator winding slot modulation operator M ss (θ) can be obtained in the same way, with the winding slots offset by half a pole pitch relative to the permanent magnet slots, and can be expressed as:

[0041]

[0042] Where v is a positive integer, N s is the number of stator teeth; a ss0 、a ssv are the DC and AC components of the Fourier coefficients respectively, and can be similarly calculated based on the stator slot parameter b s , t s , g and the intermediate variable α they constitute s , ε2 is obtained, no further details are given. The modulation operator satisfies the linear rate, and the stator modulation operator M s (θ) is the linear product of the stator winding slot modulation operator and the stator permanent magnet slot modulation operator, which can be expressed as:

[0043] Ms (θ)=M sm (θ)M ss (θ)

[0044] Figure 3 The equivalent mathematical model diagram of the rotor modulation operator is shown in Figure 2. The rotor is in a rotating state, and the rotor permanent magnet slot modulation operator M r (θ,t) can be expressed as:

[0045]

[0046] Where v is a positive integer, N r is the number of rotor poles, Ω r is the rotation speed; a r0 、a rv are the DC and AC components of the Fourier coefficients respectively. Similarly, according to the rotor permanent magnet slot parameter b r , t r , g and the intermediate variable α they constitute r , ε3 is obtained, which will not be repeated here. The modulation operator satisfies the linear rate, and the rotor modulation operator is the rotor permanent magnet slot modulation operator.

[0047] It can be seen that the modulation operator is a mathematical function with a value between 0 and 1, which represents the drop of the magnetomotive force at each slot, so that the uneven air gap caused by the stator split teeth and the rotor salient poles is equivalent to a uniform air gap.

[0048] Step 2: Establish an equivalent mathematical model of the initial stator permanent magnet magnetomotive force, the initial rotor permanent magnet magnetomotive force, and the initial armature magnetomotive force.

[0049] According to the parameters of the stator and rotor permanent magnets, the initial stator permanent magnet magnetomotive force models and the initial rotor permanent magnet magnetomotive force models are established respectively. For the convenience of modeling, the initial magnetomotive force can be simplified to be equivalent to a square wave; according to the position distribution of the stator three-phase armature winding, the initial magnetomotive force of each phase is superimposed to establish the initial armature magnetomotive force model. For the convenience of modeling, the initial magnetomotive force can be simplified to be equivalent to a square wave and decomposed into forward and reverse components.

[0050] Figure 4 For the convenience of modeling, the initial stator permanent magnet magnetomotive force F spm0 (θ) can be simplified to be equivalent to a square wave, expressed as:

[0051]

[0052] Where i is a positive integer, P s is the number of stator permanent magnet poles; F spmi is the amplitude of the Fourier decomposition of each order harmonic, expressed as:

[0053]

[0054] Among them F pm is the magnetomotive force amplitude of the permanent magnet, h m is the thickness of the stator permanent magnet, α m is the intermediate variable mentioned above, and g is the air gap length.

[0055] Figure 5 The initial rotor permanent magnet magnetomotive force mathematical model diagram is shown in Figure 2. The rotor permanent magnet is in a rotating state. For the convenience of modeling, the initial rotor permanent magnet magnetomotive force F rpm0 (θ, t) can be simplified to be equivalent to a square wave, expressed as:

[0056]

[0057] Where j is a positive integer P r is the number of rotor permanent magnet poles, Ω r is the rotation speed; F rpmj is the amplitude of the Fourier decomposition of each order harmonic, expressed as:

[0058]

[0059] Among them F pm is the magnetomotive force amplitude of the permanent magnet, h r is the thickness of the rotor permanent magnet, α r is the intermediate variable mentioned above, and g is the air gap length.

[0060] Figure 6 The mathematical model diagram of the initial armature magnetomotive force is shown in Figure 2. According to the position distribution of the three-phase armature winding of the stator, the initial magnetomotive force F of each phase is calculated. A (θ,t),F B (θ,t),F C (θ, t) can be superimposed to establish the initial armature magnetomotive force F w0 (θ, t) model, for the convenience of modeling, the initial magnetomotive force can be simplified to be equivalent to a square wave and decomposed into the positive rotation component F p (θ, t) and the inverse component F n (θ,t), expressed as:

[0061]

[0062] Where m satisfies 6r±1 and r is a non-negative integer, P w is the number of winding fundamental wave pole pairs, ω e is the electrical speed; F Фm is the amplitude of the Fourier decomposition of each order harmonic, which can be expressed as:

[0063]

[0064] Among them I mis the rated current amplitude, N c is the number of winding turns, k w is the winding coefficient.

[0065] Step 3: Based on the modulation relationship of each part, each initial magnetomotive force is multiplied by one or more corresponding modulation operators to solve the actual magnetomotive force component of each part in the air gap.

[0066] Considering the continuous-pole permanent magnet structure of the stator of the dual permanent magnet vernier motor, the modulation process of the stator permanent magnet magnetomotive force no longer needs to consider the modulation effect of the stator permanent magnet slots. The initial stator permanent magnet magnetomotive force is multiplied by the stator winding slot modulation operator to obtain the stator adjusted stator permanent magnet magnetomotive force, and the stator adjusted stator permanent magnet magnetomotive force is multiplied by the rotor modulation operator to obtain the actual stator permanent magnet magnetomotive force.

[0067] The actual magnetomotive force is obtained by modulating the initial magnetomotive force. Considering the continuous pole permanent magnet structure of the stator of the dual permanent magnet vernier motor, the modulation process of the stator permanent magnet magnetomotive force does not need to consider the modulation effect of the stator permanent magnet slot. The initial stator permanent magnet magnetomotive force is multiplied by the stator winding slot modulation operator to obtain the stator adjusted stator permanent magnet magnetomotive force. The stator adjusted stator permanent magnet magnetomotive force is multiplied by the rotor modulation operator to obtain the actual stator permanent magnet magnetomotive force F. spm (θ,t), expressed as:

[0068] F spm (θ,t)=F spm0 (θ)M ss (θ)M r (θ,t)

[0069] Considering the continuous pole permanent magnet structure of the dual permanent magnet vernier motor rotor, the modulation process of the rotor permanent magnet magnetomotive force no longer needs to consider the modulation effect of the rotor. The initial rotor permanent magnet magnetomotive force multiplied by the stator total modulation operator can directly obtain the actual rotor permanent magnet magnetomotive force F rpm (θ,t), expressed as:

[0070] F rpm (θ,t)=F rpm0 (θ,t)M s (θ)

[0071] The modulation process of the armature magnetomotive force must take into account the modulation effects of the stator and rotor at the same time. The initial armature magnetomotive force multiplied by the stator modulation operator can be obtained as the stator modulated armature magnetomotive force, and the stator modulated armature magnetomotive force multiplied by the rotor modulation operator can be obtained as the actual armature magnetomotive force F w (θ,t), expressed as:

[0072] F w (θ,t)=F w0 (θ,t)M s (θ)M r (θ,t)

[0073] Step 4: The non-uniform air gap of each component has been converted into a uniform air gap through the above process, thereby analyzing the magnetic field components of each part in the air gap.

[0074] Multiplying the actual magnetomotive force component of each part by the uniform air gap coefficient μ0 / g can obtain the magnetic field of each part. The superposition of the stator permanent magnet magnetic field and the rotor permanent magnet magnetic field can obtain the no-load air gap magnetic field, and the superposition of the no-load air gap magnetic field and the armature magnetic field can obtain the loaded air gap magnetic field; based on the analyzed magnetic field, electromagnetic parameters such as induced electromotive force and torque can be further obtained.

[0075] Figure 7 、 8 Figures 9 and 9 compare the stator permanent magnet magnetic field, rotor permanent magnet magnetic field, and armature magnetic field analyzed using finite element software and the mathematical modulation model, respectively. It can be seen that the mathematical modulation model enables fast and flexible analysis without relying on simulation software, while ensuring calculation accuracy.

[0076] In summary, the present invention is a mathematical modulation model and magnetic field analysis method for a dual permanent magnet vernier motor. First, a mathematical model of the stator and rotor modulation operators is established according to the motor object to be analyzed; then, according to the position and size parameters of the permanent magnet and armature winding, an equivalent square wave mathematical model of the initial stator permanent magnet magnetomotive force, the initial rotor permanent magnet magnetomotive force and the initial armature magnetomotive force is established respectively; secondly, according to the modulation relationship of each part, each initial magnetomotive force is multiplied by one or more modulation operators that participate in the corresponding action, and the actual magnetomotive force component of each part in the air gap is solved; at this time, the air gap is equivalent to a uniform air gap, and according to the actual magnetomotive force of the air gap, the magnetic field component of each part in the air gap is obtained to realize the magnetic field analysis of the motor; finally, the magnetic field analyzed by the mathematical modulation model is compared with the finite element to verify the feasibility and advantages of the invention. The solution provided by the present invention can provide theoretical guidance and reference for the analysis and design of dual permanent magnet vernier motors.

Claims

1. A mathematical modulation model and magnetic field analysis method for a dual permanent magnet vernier motor, characterized in that: The following steps are involved: Step 1: Establish mathematical models of stator permanent magnet slot modulation operator, stator winding slot modulation operator and rotor permanent magnet slot modulation operator; Step 2: Establish an equivalent mathematical model of the initial stator permanent magnet magnetomotive force, the initial rotor permanent magnet magnetomotive force, and the initial armature magnetomotive force; Step 3: Based on the modulation relationship of each part of the dual permanent magnet vernier motor, each initial magnetomotive force is multiplied by one or more corresponding modulation operators to solve for the actual magnetomotive force component of each part in the air gap; Step 4: The non-uniform air gap of each component has been converted into a uniform air gap after steps 1 to 3, thereby analyzing the magnetic field components of each part in the air gap; The specific process of step 1 is as follows: establishing mathematical models of a stator permanent magnet slot modulation operator, a stator winding slot modulation operator, and a rotor permanent magnet slot modulation operator; the modulation operator satisfies a linear rate, the stator modulation operator is a linear multiplication of the stator winding slot modulation operator and the stator permanent magnet slot modulation operator, and the rotor modulation operator is a rotor permanent magnet slot modulation operator; the modulation operator is a mathematical function with a value between 0 and 1, which characterizes the drop of the magnetomotive force at each slot, so that the uneven air gap caused by the stator split teeth and the rotor salient poles is equivalent to a uniform air gap; Stator permanent magnet slot modulation operator M sm (θ) is expressed as: Where v is a positive integer, P s is the number of stator permanent magnet slots; a sm0 、a smv They are the DC and AC components of the Fourier coefficients respectively, according to the stator permanent magnet slot parameter b m , t m , g and the intermediate variable α they constitute m , ε1 is obtained, which is expressed as: a sm0 =1-1.6ε1a m , Stator winding slot modulation operator M ss (θ) is obtained by the same method, with the winding slot offset by half a pole pitch relative to the permanent magnet slot, expressed as: Where v is a positive integer, N s is the number of stator teeth; a ss0 、a ssv They are the DC and AC components of the Fourier coefficients respectively, according to the stator slot parameter b s , t s , g and the intermediate variable α they constitute s , ε2 is obtained; the modulation operator satisfies the linear rate, the stator modulation operator M s (θ) is the linear product of the stator winding slot modulation operator and the stator permanent magnet slot modulation operator, which is expressed as: M s (θ)=M sm (i)M ss (i) The rotor is in a rotating state, and the rotor permanent magnet slot modulation operator M r (θ,t) is expressed as: Where v is a positive integer, N r is the number of rotor poles, Ω r is the rotation speed; a r0 、a rv are the DC and AC components of the Fourier coefficients respectively, according to the rotor permanent magnet slot parameter b r , t r , g and the intermediate variable α they constitute r , ε3 is obtained; the modulation operator satisfies the linear rate, and the rotor modulation operator is the rotor permanent magnet slot modulation operator; The specific process of step 2 is as follows: according to the parameters of the stator and rotor permanent magnets, initial stator permanent magnet magnetomotive force models and initial rotor permanent magnet magnetomotive force models are respectively established. For the convenience of modeling, the initial magnetomotive force is simplified to be equivalent to a square wave; according to the position distribution of the stator three-phase armature winding, the initial magnetomotive force of each phase is superimposed to establish an initial armature magnetomotive force model. For the convenience of modeling, the initial magnetomotive force is simplified to be equivalent to a square wave and decomposed into a forward component and a reverse component; The initial stator permanent magnet magnetomotive force F spm0 (θ) is simplified to be equivalent to a square wave, expressed as: Where i is a positive integer, P s is the number of stator permanent magnet poles; F spmi is the amplitude of the Fourier decomposition of each order harmonic, expressed as: Among them F pm is the magnetomotive force amplitude of the permanent magnet, h m is the thickness of the stator permanent magnet, α m is the above intermediate variable, g is the air gap length; The rotor permanent magnet is in a rotating state. To facilitate modeling, the initial rotor permanent magnet magnetomotive force F rpm0 (θ, t) is simplified to be equivalent to a square wave, expressed as: Where j is a positive integer P r is the number of rotor permanent magnet poles, Ω r is the rotation speed; F rpmj is the amplitude of the Fourier decomposition of each order harmonic, expressed as: Among them F pm is the magnetomotive force amplitude of the permanent magnet, h r is the thickness of the rotor permanent magnet, α r is the above intermediate variable, g is the air gap length; According to the position distribution of the stator three-phase armature winding, the initial magnetomotive force F of each phase is A (θ,t),F B (θ,t),F C (θ, t) are superimposed to establish the initial armature magnetomotive force F w0 (θ, t) model, for the convenience of modeling, the initial magnetomotive force is simplified to be equivalent to a square wave and decomposed into the positive rotation component F p (θ, t) and the inverse component F n (θ,t), expressed as: Where m satisfies 6r±1 and r is a non-negative integer, P w is the number of winding fundamental wave pole pairs, ω e is the electrical speed; F Фm is the amplitude of the Fourier decomposition of each order harmonic, expressed as: Among them I m is the rated current amplitude, N c is the number of winding turns, k w is the winding coefficient.

2. The mathematical modulation model and magnetic field analysis method of a dual permanent magnet vernier motor according to claim 1 is characterized in that: The selected dual permanent magnet vernier motor has a stator with 12 slots, a stator permanent magnet with 12 poles, and a rotor permanent magnet with 19 poles. It includes six parts: stator core, stator permanent magnet, rotor core, rotor permanent magnet, stator winding, and air gap. The stator core contains 12 split teeth, each split tooth is split into 2 small teeth, which participate in modulation. The 12 continuous-pole permanent magnets of the stator are respectively embedded in the slots produced by each split tooth. The rotor core contains 19 salient poles, which participate in modulation. The 19 continuous-pole permanent magnets of the rotor are respectively embedded in the slots produced by the salient poles. The stator armature winding adopts fractional slot concentrated winding. The air gap is located between the stator and the rotor, and the air gap thickness is 0.7mm.

3. The mathematical modulation model and magnetic field analysis method of a dual permanent magnet vernier motor according to claim 1 is characterized in that: In step 3, considering the continuous-pole permanent magnet structure of the stator of the dual permanent magnet vernier motor, the modulation process of the stator permanent magnet magnetomotive force no longer needs to consider the modulation effect of the stator permanent magnet slots. The initial stator permanent magnet magnetomotive force is multiplied by the stator winding slot modulation operator to obtain the stator adjusted stator permanent magnet magnetomotive force, and the stator adjusted stator permanent magnet magnetomotive force is multiplied by the rotor modulation operator to obtain the actual stator permanent magnet magnetomotive force.

4. The mathematical modulation model and magnetic field analysis method of a dual permanent magnet vernier motor according to claim 1, characterized in that: In step 3, considering the continuous pole permanent magnet structure of the dual permanent magnet vernier motor rotor, the modulation process of the rotor permanent magnet magnetomotive force no longer needs to consider the modulation effect of the rotor. The initial rotor permanent magnet magnetomotive force multiplied by the stator total modulation operator can directly obtain the actual rotor permanent magnet magnetomotive force.

5. The mathematical modulation model and magnetic field analysis method of a dual permanent magnet vernier motor according to claim 1 is characterized in that: In step 3, the modulation process of the armature magnetomotive force must take into account the modulation effects of the stator and the rotor at the same time. The initial armature magnetomotive force multiplied by the stator modulation operator can obtain the stator modulated armature magnetomotive force, and the stator modulated armature magnetomotive force multiplied by the rotor modulation operator can obtain the actual armature magnetomotive force.

6. The mathematical modulation model and magnetic field analysis method of a dual permanent magnet vernier motor according to claim 1, characterized in that: The specific process of step 4 is as follows: the actual magnetomotive force component of each part is multiplied by the uniform air gap coefficient μ0 / g to obtain the magnetic field of each part, the stator permanent magnet magnetic field and the rotor permanent magnet magnetic field are superimposed to obtain the no-load air gap magnetic field, and the no-load air gap magnetic field and the armature magnetic field are superimposed to obtain the loaded air gap magnetic field; based on the analyzed magnetic field, electromagnetic parameters such as induced electromotive force and torque can be obtained.

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