Modeling method for magnetic network model of self-starting permanent magnet synchronous motor

Abstract

The invention provides an improved permanent magnet synchronous motor magnetic network modeling method, relates to the field of electromagnetic calculation of motors, and is suitable for design and calculation of electromagnetic characteristics of a self-starting permanent magnet synchronous motor. According to the structural characteristics and magnetic distribution of the permanent magnet synchronous motor, a magnetic network model of a stator, a rotor and an air gap is built, and rotor conducting bar magnetic leakage, a rotor conducting bar skin effect and non-uniform saturation of a magnetic bridge area between a rotor starting cage and a permanent magnet in the starting process and under a steady-state working condition are mainly considered; the invention provides a modeling method for a rotor conducting bar area and a magnetic bridge area, so that the magnetic flux of the rotor conducting bar area and the magnetic bridge area can be uniformly distributed in a unit, the local magnetic resistance is effectively reduced, and the magnetic passing degree concentration is avoided. By means of the method, electromagnetic analysis under the steady-state working condition can be conducted, the electromagnetic performance in the starting process can be analyzed according to the characteristic that the motor requires the large starting torque, the electromagnetic performance of the motor under different working conditions is analyzed more accurately, and therefore design and calculation of the electromagnetic performance of the motor are better conducted.

Description

Technical Field

[0001] This invention belongs to the field of electromagnetic field calculation for electric motors, and relates to a method for modeling the magnetic network of a self-starting permanent magnet synchronous motor. Technical Background

[0002] Self-starting permanent magnet synchronous motors (PMSMs) exhibit high operating performance, primarily characterized by high torque density, a wide constant power operating range, and low permanent magnet eddy current losses. PMSMs are favored due to their relatively simple structure, high reliability, and stable operation. The rotor of a self-starting PMSM has a unique cage-like rotor slot structure, resulting in a certain skin effect. Furthermore, the rotor contains magnetic bridges, and the magnetic bridge region exhibits non-uniform saturation. The main reason for the uneven magnetic flux density distribution in the magnetic bridge region is the special structure of the rotor starting cage in the self-starting PMSM. As one approaches the center of the magnetic bridge region, the cross-sectional area of ​​the magnetic flux path decreases, forcing the magnetic lines of force to concentrate, causing a rapid increase in magnetic flux per unit area and a significant increase in magnetic flux density, thus leading to the non-uniform saturation phenomenon. The calculation of the magnetic flux density in the magnetic bridge region and the rotor starting cage is complex; therefore, precise partitioning of the rotor magnetic network is necessary to meet the electromagnetic analysis and calculation requirements in the early stages of self-starting PMSM design.

[0003] Current methods for analyzing the magnetic field of permanent magnet synchronous motors mainly include the finite element method (FEM), analytical methods, and equivalent magnetic network methods. Among these, the FEM offers high accuracy and visualization of results, but its drawbacks are significant: long solution time, low efficiency, and high computer resource consumption. Therefore, the FEM is primarily used for later optimization analyses. Analytical methods describe the magnetic field distribution using mathematical formulas, offering fast calculation speeds but relatively poor accuracy, thus failing to meet the demands of electromagnetic analysis. The equivalent magnetic network method, based on the principle of equivalent flux tubes, considers the motor as a whole composed of multiple unit parts, equating each unit to a magnetic permeability. The equivalent permeability within each unit region is identical. These equivalent permeabilities are then connected to form a network. The nodal magnetic potentials and unit region magnetic flux density of the entire magnetic network are calculated using Kirchhoff's voltage law and current law, thereby obtaining the electromagnetic characteristics of the entire motor. This method offers high accuracy and fast solution speed. Summary of the Invention

[0004] The purpose of this invention is to improve the magnetic network modeling of self-starting permanent magnet synchronous motors. The equivalent magnetic network method balances computational efficiency and accuracy by representing the motor's magnetic field as a network of magnetic permeability elements, but it has shortcomings in handling the nonlinear permeability of ferromagnetic materials and the non-uniform saturation of magnetic bridges. To address these shortcomings and establish a more accurate rotor magnetic network model, this patent focuses on rotor bar leakage flux, the skin effect of rotor bars, and the non-uniform saturation of the magnetic bridge region between the starting cage and the permanent magnet during startup and steady-state operation. Addressing the magnetic flux density distribution problem, a magnetic bridge region modeling method and a rotor cage bar region modeling method are proposed. This method introduces a cross-shaped magnetic permeability. This cross-shaped permeability allows for a uniform distribution of magnetic flux in the magnetic bridge region and rotor cage bars within the unit cell, while effectively reducing local magnetic reluctance and avoiding magnetic flux concentration, thereby obtaining better electromagnetic characteristics of the motor.

[0005] To achieve the above objectives, the technical solution proposed in this invention is: a method for modeling the magnetic network of a self-starting permanent magnet synchronous motor, comprising the following steps:

[0006] Step 1: Perform two-dimensional finite element simulation of the permanent magnet synchronous motor based on its structural dimensions, materials and other parameters. According to the magnetic field distribution, divide the motor into regions, specifically the stator region, air gap and rotor region. Then, convert each part into the corresponding magnetic permeability.

[0007] Step 2: Establish the stator magnetic network model of the permanent magnet synchronous motor;

[0008] Step 3: Establish the air gap magnetic network model of the permanent magnet synchronous motor;

[0009] Step 4: Establish the rotor magnetic network model of the permanent magnet synchronous motor.

[0010] Step 5: Write the nodal magnetic potential equations based on the equivalent magnetic network model of each part and solve them iteratively.

[0011] Furthermore, in step 2, the motor stator magnetic network model is mainly divided into magnetic network models of the stator teeth, stator yoke, and stator slots. Considering the uneven magnetic flux distribution in the motor yoke and the fact that the area of ​​the stator yoke is larger than that of the teeth, to improve the accuracy of the motor's electromagnetic performance analysis, this invention divides the stator yoke radially and tangentially when modeling the yoke magnetic network, equivalent to a single layer of radial magnetic permeability G. sr Two layers of tangential magnetic permeability G st This equivalent method allows for a more precise analysis of the stator yoke magnetic flux density distribution, expressed as:

[0012]

[0013] Where l is the axial length of the motor; μ is the permeability of the core material; h1 and h2 are the inner and outer diameters of the sector magnetic permeability corresponding to a single stator slot; R1 and R2 are the inner and outer radii of the stator yoke; θ is the angle corresponding to a single stator slot; and h is the tooth height.

[0014] The stator slot has a large internal area and is not close to the air gap, so the magnetic flux is very small. However, considering the influence of leakage flux, the stator slot leakage flux is equivalent to 5 magnetic inductances.

[0015] The tangential component of the stator tooth flux is relatively small, so the tangential magnetic permeability of the tooth is ignored, and only the radial magnetic permeability of the tooth is retained. The stator winding is equivalent to a magnetomotive force source.

[0016] Furthermore, in step 3, in the motor air gap magnetic network model, the air gap permeability is determined by the relative positions of the stator and rotor tooth slots, and the air gap structure is as follows: Figure 1 As shown. When the stator and rotor teeth are completely misaligned, the permeability approaches 0, at which point it is approximately non-existent in terms of air gap permeability. When the stator and rotor teeth begin to intersect, the air gap permeability G changes with the angle between the centerlines of the stator and rotor teeth. i,j In [0, G max Within a certain range, the stator and rotor teeth are completely opposite, meaning the maximum value G is reached when the larger tooth contains the smaller tooth. max Therefore, the air gap permeability is constantly changing, and the formula for the air gap permeability value is:

[0017]

[0018] The expression for some parameters in the above formula is as follows:

[0019]

[0020] In the formula, γ, γ t '、γ t Angular variable is used to divide different intervals to determine the formula for calculating air gap permeability within each interval. N st N is the stator tooth tip width; rt M is the rotor tooth tip width; ss M is the width of the stator slot; rs N is the rotor slot width; tmin δ is the smaller value of the stator and rotor tooth width; δ is the air gap length; D si D is the inner diameter of the stator. ro D is the rotor outer diameter; ag It is the arithmetic average of the stator inner diameter and the rotor outer diameter.

[0021] Further, in step 4, the motor rotor magnetic network model is mainly divided into magnetic network models of the permanent magnet region, magnetic bridge, rotor teeth, rotor yoke, and rotor bars. A cross-shaped magnetic guide is established on the upper part of the permanent magnet; leakage flux is considered and leakage magnetic guide is established for the permanent magnet magnetic bridge; a pair of permanent magnets is equivalent to four magnetic guides and magnetomotive force sources; the radial component of the magnetic flux in the rotor yoke is relatively small, so it is equivalent to two tangential magnetic guides and one radial magnetic guide near the rotor shaft; the skin effect of the rotor bars is considered in the rotor teeth, and they are equivalent to cross-shaped magnetic guides and magnetomotive force sources; the bars in the rotor starting cage are considered for leakage flux and skin effect during starting and steady-state operation. It must be mentioned that the skin effect can improve starting torque and reduce starting current to a certain extent, thereby improving the starting and running performance of the motor. The slot shape of the rotor slots is closely related to this; to a certain extent, square and deep slots can provide greater starting torque, while circular and shallow slots are relatively weaker. Therefore, when using a square deep slot, the leakage flux inside the slot can be ignored due to the skin effect. At this time, no magnetic conduction will be established in the slot during the startup process and steady-state operation. When using a circular shallow slot, the skin effect is weaker and the leakage flux inside the slot is larger. During the startup process and steady-state operation, it is equivalent to a cross-shaped magnetic conduction. The main reason for the uneven magnetic flux density distribution in the permanent magnet bridge region is the starting cage structure of the self-starting motor rotor. As it moves closer to the center of the magnetic bridge region, the cross-sectional area of ​​the magnetic flux path decreases in the central region, and the magnetic lines of force are forced to concentrate, causing the magnetic flux per unit area to rise rapidly and the magnetic flux density to increase significantly. This leads to the phenomenon of uneven magnetic flux density distribution and saturation. Therefore, it is equivalent to five cross-shaped magnetic conductions and one rectangular magnetic conduction, so that the magnetic flux in the magnetic bridge region can be evenly distributed within the unit.

[0022] Furthermore, in step 5, the nodal magnetic potential equation is expressed as:

[0023]

[0024] In the formula, n is the number of nodes, F n×1 Let Φ be the nodal magnetomotive force matrix. n×1 G is the total magnetic flux matrix flowing into the node. n×n Let be the nodal permeability matrix, and its expression is:

[0025]

[0026] Wherein, G(i,i) is the self-derivative of node i, and its value is always positive; G(i,j) is the mutual derivative between nodes i and j, and its value is always negative, but for nodes i and j that are not connected, G(i,j) is 0.

[0027] The beneficial effects of this invention are:

[0028] This invention focuses on the leakage flux of rotor bars, the skin effect of rotor slots, and the non-uniform saturation of the magnetic bridge region between the starting cage and the permanent magnet during startup and steady-state operation. To address the magnetic flux density distribution problem, a magnetic bridge region modeling method and a rotor cage bar region modeling method are proposed. This method introduces a cross-shaped magnetic permeability. This cross-shaped permeability allows for a uniform distribution of magnetic flux in the magnetic bridge region and rotor cage bars within the unit cell, improving the computational accuracy of the magnetic network model. Simultaneously, it effectively reduces local magnetic reluctance and avoids magnetic flux concentration, thereby obtaining better electromagnetic characteristics of the motor and meeting the electromagnetic analysis and calculation needs in the early stages of motor design. Attached Figure Description

[0029] Figure 1 This is a diagram of the air gap structure of the motor according to an embodiment of the present invention;

[0030] Figure 2 This is an equivalent model diagram of the unipolar magnetic network of the motor in an embodiment of the present invention; Detailed Implementation

[0031] To make the purpose, technical solution, and advantages of this invention patent clearer, the technical solutions in the embodiments of this invention will be clearly and completely described below with reference to the accompanying drawings.

[0032] Based on the structural characteristics of self-starting permanent magnet synchronous motors, this invention uses the equivalent magnetic network method to establish a magnetic network model for the motor.

[0033] This embodiment uses a specific self-starting permanent magnet synchronous motor as an example. The embodiment is an 8-pole self-starting permanent magnet synchronous motor with 48 slots on the outer stator and 40 slots on the inner rotor. The rotor and stator slot types are as follows: Figure 1 As shown, the magnetic network model is divided into three parts: stator, rotor, and air gap. The stator part includes stator yoke, stator teeth, and stator slots, while the rotor part includes rotor yoke, rotor teeth, rotor slots, permanent magnets, and magnetic bridges.

[0034] The establishment and solution of the magnetic network are accomplished through the following steps:

[0035] Step 1: Perform two-dimensional finite element simulation of the permanent magnet synchronous motor based on its structural dimensions, materials, and other parameters. Divide the motor into regions based on the simulated magnetic lines of force, and then convert each part into the corresponding magnetic permeability.

[0036] Furthermore, in step 1, the region is divided into three parts: stator, rotor, and air gap. Each region is divided into grids, and magnetic network modeling is performed based on the distribution of magnetic lines of force of each component.

[0037] Step 2: Establish the stator magnetic network model of the permanent magnet synchronous motor. The stator region is mainly divided into the magnetic network models of the stator teeth, stator yoke, and stator slots. Considering the uneven magnetic flux distribution in the motor yoke and the fact that the area of ​​the stator yoke is larger than that of the teeth, to improve the accuracy of the motor's electromagnetic performance analysis, this invention divides the stator yoke radially and tangentially when modeling the yoke magnetic network, equivalent to a single layer of radial magnetic permeability G. sr Two layers of tangential magnetic permeability G st This equivalent method allows for a more precise analysis of the stator yoke magnetic flux density distribution, expressed as:

[0038]

[0039] Where l is the axial length of the motor; μ is the permeability of the core material; h1 and h2 are the inner and outer diameters of the sector magnetic permeability corresponding to a single stator slot; R1 and R2 are the inner and outer radii of the stator yoke; θ is the angle corresponding to a single stator slot; and h is the tooth height.

[0040] The stator slots have a large internal area and are not close to the air gap, resulting in a very small magnetic flux. However, considering the influence of leakage flux, the stator slot leakage flux is equivalent to 5 magnetic permeabilities. The tangential component of the stator tooth flux is relatively small, so the tooth tangential flux is ignored, and only the tooth radial flux is retained. The stator windings are equivalent to a magnetomotive force source. The stator magnetic network model is as follows: Figure 2 As shown.

[0041] Step 3: Establish the air gap magnetic network model of the permanent magnet synchronous motor. The air gap permeability is determined by the relative position of the stator and rotor teeth. When the stator and rotor teeth are completely misaligned, the permeability approaches 0, and at this point, it is approximately as if there is no air gap permeability. When the stator and rotor teeth begin to intersect, the permeability G changes with the angle between the center lines of the stator and rotor teeth. i,j In [0, G max Within a certain range, the stator and rotor teeth are completely opposite, meaning the maximum value G is reached when the larger tooth contains the smaller tooth. max Therefore, the air gap permeability is constantly changing.

[0042] The self-starting permanent magnet synchronous motor studied in this embodiment has 6 stator teeth and 5 rotor teeth per pole pair, for a total of 48 stator teeth and 40 rotor teeth. The initial time t0 is set when the rotor has not yet rotated, and the position angle matrix of the stator teeth is set to γ. s1 =[j1,j2…j 48 The position angle matrix of the rotor teeth is set to γ. r1 = [k1,k2…k 40 Using the center line of the stator and rotor teeth as a reference, the relative position angle matrix γ between the stator and rotor can be formed. sr1 This matrix is ​​a 48×40 matrix. Let the mechanical angular velocity of the rotor be ω. mIf we set the time interval between any two adjacent records to Δt, then the corresponding change in the rotor tooth position angle matrix is ​​γ. r2 =[k1+ω m ·Δt,k2+ω m ·Δt…k 40 +ω m ·Δt], thus forming a new stator-rotor relative position angle matrix γ sr2 Therefore, the air gap permeability matrix between the stator and rotor teeth can be obtained as follows:

[0043]

[0044] In the above formula, G s,r This represents the magnetic permeability between the stator and rotor teeth at the corresponding moment. An air gap magnetic network model is established as follows: Figure 2 As shown.

[0045] Step 4: Establish the rotor magnetic network model of the permanent magnet synchronous motor. The rotor region is mainly divided into the permanent magnet region, magnetic bridge, rotor teeth, rotor yoke, and rotor bars. A cross-shaped magnetic flux is established on the upper part of the permanent magnet; leakage flux is considered and leakage flux is established for the permanent magnet magnetic bridge; a pair of permanent magnets is equivalent to four magnetic fluxes and a magnetomotive force source; the radial component of the magnetic flux in the rotor yoke is relatively small, so it is equivalent to two tangential magnetic fluxes and one radial magnetic flux near the shaft; the skin effect of the rotor bars is considered in the rotor teeth, and they are equivalent to a cross-shaped magnetic flux and a magnetomotive force source; the leakage flux and skin effect of the bars in the rotor starting cage are considered during startup and under steady-state conditions. Since the motor in this embodiment has shallow circular slots, the leakage flux in the slots is relatively large. During startup and under steady-state conditions, it is equivalent to a cross-shaped magnetic inductance. The main reason for the uneven magnetic flux density distribution in the permanent magnet bridge region is the rotor starting cage structure of the self-starting motor. As it approaches the center of the magnetic bridge region, the cross-sectional area of ​​the magnetic flux path decreases in the central region, forcing the magnetic lines of force to concentrate, causing the magnetic flux per unit area to rise rapidly and the magnetic flux density to increase significantly, thus triggering the phenomenon of uneven magnetic flux density distribution saturation. Therefore, it is equivalent to five cross-shaped magnetic inductances and one rectangular magnetic inductance, so that the magnetic flux in the magnetic bridge region can be evenly distributed within the unit. The rotor magnetic network model is as follows: Figure 2 As shown.

[0046] Step 5: Based on the equivalent magnetic network model of each part, derive the nodal magnetic potential equations and solve them iteratively. The nodal magnetic permeability matrix is ​​expressed as:

[0047]

[0048] Wherein, G(i,i) is the self-derivative of node i, and its value is always positive; G(i,j) is the mutual derivative between nodes i and j, and its value is always negative, but for nodes i and j that are not connected, G(i,j) is 0.

[0049] Furthermore, the nodal magnetic potential matrix is ​​solved based on the nodal permeability matrix and the nodal magnetic flux matrix. The equation for the nodal magnetic potential matrix is ​​as follows:

[0050]

[0051] In the formula, n is the number of nodes, F n×1 Let Φ be the nodal magnetomotive force matrix. n×1 Let be the total magnetic flux matrix flowing into the node.

[0052] Furthermore, the solution mainly involves solving a system of nonlinear equations. In this embodiment, the super-relaxation iterative method (SOR) is chosen for the solution. Compared with other methods, the SOR iteration increases the relaxation factor, which greatly increases the convergence speed. The following equation is the computational component form for solving the nodal magnetic potential equation:

[0053]

[0054] The above formula can be simplified to:

[0055] F (k+1) =(D-wL) -1 [(1-w)D+wU]F (k) +w(D-wL)φ

[0056] In the formula, k is the iteration number, and w is the over-relaxation factor (1). <w<2)。

[0057] Furthermore, when the iteration converges or reaches the upper limit of convergence, the iteration process terminates, and the solution is completed. That is, it conforms to the following equation:

[0058] |F (k) -F (k-1) |<ε a

[0059] In the formula, ε a To ensure accuracy, the larger the error, the worse the accuracy; therefore, the accuracy should not be too large.

[0060] In summary, this invention proposes a magnetic network modeling method for a self-starting permanent magnet synchronous motor, which ensures accurate and efficient analysis of the motor's performance under conditions that fully consider the rotor bar leakage flux, rotor bar skin effect, and non-uniform saturation of the magnetic bridge region between the rotor starting cage and the permanent magnet during startup and steady-state operation.

[0061] The above describes preferred embodiments of the present invention, but the scope of application of the present invention is not limited to the above embodiments. All derivative technical solutions based on the concept of the present invention fall within the protection scope of the present invention. Any equivalent modifications and refinements made by other skilled in the art without departing from the principles of the present invention are within the protection scope of the present invention and are protected by the appended claims.

Claims

1. A method for modeling a magnetic network of a self-starting permanent magnet synchronous motor, comprising the following steps: Step 1: Perform two-dimensional finite element simulation of the permanent magnet synchronous motor based on its structural dimensions, materials and other parameters. Based on the simulated magnetic lines of force, divide the motor into regions, specifically the stator region, air gap and rotor region. Then, convert each part into the corresponding magnetic permeability. Step 2: Establish the stator magnetic network model of the permanent magnet synchronous motor; Step 3: Establish the air gap magnetic network model of the permanent magnet synchronous motor; Step 4: Establish the rotor magnetic network model of the permanent magnet synchronous motor; Step 5: Based on the equivalent magnetic network model of each part, list the nodal magnetic potential equations and solve them iteratively.

2. The modeling method for the magnetic network model of a self-starting permanent magnet synchronous motor according to claim 1, wherein step 2 is characterized in that, The specific method for establishing the stator magnetic network model is as follows: The stator magnetic network model of an electric motor is mainly divided into magnetic network models of the stator teeth, stator yoke, and stator slots. Considering the uneven magnetic flux distribution in the motor yoke and the fact that the area of ​​the stator yoke is larger than that of the teeth, in order to improve the accuracy of the electromagnetic performance analysis of the motor, this invention divides the stator yoke along the radial and tangential directions when modeling the magnetic network of the yoke, which is equivalent to one layer of radial magnetic permeability and two layers of tangential magnetic permeability. The stator slot has a large internal area and is not close to the air gap, so the magnetic flux is very small. However, considering the influence of leakage flux, the stator slot leakage flux is equivalent to 5 magnetic inductances. The tangential component of the stator tooth flux is relatively small, so the tangential magnetic permeability of the tooth is ignored, and only the radial magnetic permeability of the tooth is retained. The stator winding is equivalent to a magnetomotive force source.

3. The modeling and method for the magnetic network model of a self-starting permanent magnet synchronous motor according to claim 1, wherein step 3 is characterized in that... The special structure between the stator slots and the rotor starting cage of the motor; Establishing an air gap magnetic network model requires considering that: the air gap permeability is determined by the relative positions of the stator and rotor teeth. When the stator and rotor teeth are completely misaligned, the permeability approaches 0, which is approximately equivalent to no air gap permeability. When the stator and rotor teeth begin to intersect, the permeability changes with the angle between the centerlines of the stator and rotor teeth in the range [0, G]. max Within a certain range, the magnetic permeability reaches its maximum value G when the stator and rotor teeth are completely opposite, that is, when the larger tooth contains the smaller tooth. max Therefore, it is important to consider that the magnetic permeability in the air gap is constantly changing.

4. The method for modeling the magnetic network of a self-starting permanent magnet synchronous motor according to claim 1, wherein step 4 is characterized in that the specific method for establishing the rotor magnetic network model is as follows: The magnetic network model of the motor rotor is mainly divided into the magnetic network models of the permanent magnet region, magnetic bridge, rotor teeth, rotor yoke and rotor bars. A cross-shaped magnetic permeability is established on the upper part of the permanent magnet; Permanent magnets should consider leakage flux and establish leakage flux permeability; A permanent magnet is equivalent to two magnetic permeability sources and two magnetic potential sources. The rotor teeth take into account the skin effect of the rotor bars and are equivalent to a cross-shaped magnetic permeability; The radial component of the magnetic flux in the rotor yoke is relatively small, so in the part of the rotor yoke near the shaft, it is equivalent to two tangential magnetic permeability and one radial magnetic permeability. Considering the leakage flux of the guide bars in the rotor starting cage during the starting process and under steady-state conditions, they are equivalent to cross-shaped magnetic permeability. The rotor teeth take into account the skin effect of the rotor slots during startup and the magnetic field distribution under steady-state conditions. During startup, they are equivalent to a magnetomotive force source and a cross-shaped magnetic permeability, and under steady-state conditions, they are equivalent to a cross-shaped magnetic permeability. The main reason for the uneven magnetic flux density distribution in the permanent magnet bridge region is the special structure of the rotor starting cage of the self-starting motor. As it moves closer to the center of the bridge region, the cross-sectional area of ​​the magnetic flux path decreases in the central region, and the magnetic lines of force are forced to concentrate, causing the magnetic flux per unit area to rise rapidly and the magnetic flux density to increase significantly. This leads to the phenomenon of uneven magnetic flux density distribution and saturation. Therefore, it is equivalent to five cross-shaped magnetic inductances and one rectangular magnetic inductance, so that the magnetic flux in the bridge region can be evenly distributed within the unit.

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