A 3D Eddy Current Loss Analytical Calculation Method for Surface Mounted Permanent Magnet Motor Rotors

Abstract

The present invention discloses a three-dimensional eddy current loss analytical calculation method for the rotor of a surface-mounted permanent magnet motor, which relates to the technical field of eddy current loss calculation of the rotor of a permanent magnet motor. The present invention includes the following steps: S1, analytical calculation of the magnetic field of the surface-mounted permanent magnet motor; S2, extending the magnetic field obtained by the analytical calculation to the axial length, and calculating the axial component and the tangential component of the three-dimensional eddy current density of the rotor; S3, establishing a three-dimensional eddy current loss analytical calculation model according to the axial component and the tangential component of the three-dimensional eddy current density; S4, obtaining the three-dimensional eddy current loss of the rotor of the surface-mounted permanent magnet motor according to the three-dimensional eddy current loss analytical calculation model established in S3. The present invention can more accurately simulate and predict the three-dimensional eddy current situation inside the rotor, provide more real and accurate three-dimensional eddy current loss analytical calculation results, and is of great help in optimizing and reducing the three-dimensional eddy current loss of the rotor of the permanent magnet motor, enabling the permanent magnet motor to operate within a safe range and improving its electromagnetic and thermal performance.

Description

Technical Field

[0001] The present invention relates to the technical field of calculating eddy current losses in permanent magnet motor rotors, and particularly to an analytical calculation method for three-dimensional eddy current losses in the rotor of a surface-mounted permanent magnet motor. Background Art

[0002] Permanent magnet motors have the advantages of simple structure, high torque density, and high efficiency. The application of permanent magnet motors in various industries is becoming increasingly popular, such as transportation, wind power generation systems, industry, aerospace, etc. Permanent magnet motors are excited by permanent magnets with high remanence and coercivity (such as NdFeB magnets). However, such permanent magnets are very sensitive to temperature, and their remanence decreases severely with the increase of temperature. At the same time, such permanent magnets have non-negligible conductivity, and eddy currents will also be generated during the operation of permanent magnet motors. Compared with stator iron losses and copper losses, the eddy current losses in the rotor are usually quite low. However, the heat transfer ability of the entire permanent magnet in the air gap is poor, and the influence of rotor eddy current losses cannot be ignored. In severe cases, irreversible demagnetization of the permanent magnet will occur. Calculating the eddy current losses in the rotor of a permanent magnet motor is an important part of the thermal calculation and thermal management of permanent magnet motors. At the same time, the analytical calculation of the eddy current losses in the rotor of a permanent magnet motor is beneficial to the rapid optimization design of the motor.

[0003] Chinese patent document with publication number CN116362148A and publication date of June 30, 2023 discloses a method for calculating eddy current losses in the rotor of a high-speed permanent magnet motor. The invention includes the following steps: S1, establishing an analytical model of the motor, where the analytical model of the motor includes a shaft region, a permanent magnet region, a sheath region, an air gap region, and current sheets; S2, in the polar coordinate system, obtaining the vector magnetic potential of the shaft region, the permanent magnet region, the sheath region, and the air gap region according to the Laplace equation and the complex eddy current equation; S3, solving the magnetic field function using boundary conditions; S4, calculating the eddy current losses of each harmonic; S5, performing coefficient correction; S6, calculating the total eddy current losses of the rotor; Thus, the eddy current losses of the rotor of a surface-mounted high-speed permanent magnet motor with a metal sheath for any stator current waveform can be calculated. The non-uniform three-dimensional distribution of eddy currents is considered in the calculation, and the eddy current losses of each harmonic are corrected, so that the calculation result of the eddy current losses in the rotor of the high-speed permanent magnet motor reaches a higher accuracy.

[0004] However, in the above technology, on the one hand, the spatio-temporal vector magnetic potential directly gives the three-dimensional spatio-temporal expression without considering the boundary condition that the eddy current density at the end of the permanent magnet or the rotor sheath is 0; on the other hand, the armature reaction is equivalent to an equivalent assumption of the current sheet at the slot opening, and the correction will reduce the solution accuracy, and the calculation accuracy is not good enough. Summary of the Invention

[0005] In order to overcome the defects and deficiencies existing in the above-mentioned prior art, the present invention provides an analytical calculation method for three-dimensional eddy current losses of a surface-mounted permanent magnet motor rotor, which can more accurately simulate and predict the three-dimensional eddy current situation inside the rotor, provide more realistic and accurate analytical calculation results of three-dimensional eddy current losses, and is of great help in optimizing and reducing the three-dimensional eddy current losses of the permanent magnet motor rotor, enabling the permanent magnet motor to operate within a safe range and improving its electromagnetic and thermal performance.

[0006] In order to solve the problems existing in the above-mentioned prior art, the present invention is realized through the following technical solutions.

[0007] The present invention provides an analytical calculation method for three-dimensional eddy current losses of a surface-mounted permanent magnet motor rotor, and the method includes the following steps:

[0008] S1. Analysis of the magnetic field of the surface-mounted permanent magnet motor;

[0009] S2. Extend the magnetic field obtained by analytical calculation to the axial length, and calculate the axial component and tangential component of the three-dimensional eddy current density of the rotor;

[0010] S3. According to the axial component and tangential component of the three-dimensional eddy current density, combined with the integral form of calculating the resistance loss after passing current through the resistance, establish an analytical calculation model for three-dimensional eddy current losses;

[0011] S4. Obtain the three-dimensional eddy current losses of the surface-mounted permanent magnet motor rotor according to the analytical calculation model for three-dimensional eddy current losses established in S3.

[0012] The analysis of the magnetic field of the surface-mounted permanent magnet motor includes:

[0013] Obtain the equivalent surface current of the permanent magnet;

[0014] Establish a sub-domain model of a surface-mounted high-speed permanent magnet motor with a pair of coils in the air gap and armature current in the slots; divide it into three sub-domains, calculate the magnetic fields of their corresponding sub-domains, and then obtain the radial magnetic density and tangential magnetic density of a pair of coils in the permanent magnet by combining the magnetic field calculation results of the three sub-domains;

[0015] Use the superposition principle to calculate the magnetic field and obtain the radial magnetic density and tangential magnetic density throughout the permanent magnet;

[0016] The equivalent surface current of the permanent magnet is obtained by the following formula:

[0017] J PM = H cj sinα (1)

[0018] In formula (1), H cj is the coercivity of the permanent magnet, and α is the angle between the magnetization direction and the normal direction of the permanent magnet surface.

[0019] The three sub-domains include the first type of sub-domain i between the rotor core and the inner circle of the stator core, the second type of sub-domain 2i in the stator slot opening, and the third type of sub-domain 3i in the stator slot.

[0020] The calculation of the magnetic field in the first type of sub-domain includes:

[0021] In the first type of sub-domain, there is only a Z-axis component, and the vector magnetic potential that satisfies the Laplace equation:

[0022]

[0023] In Equation (2), r is the radius of any point in the polar coordinate system of the first type of sub-domain, θ is the angle from any point to the 0° median line, and A z1 is the vector magnetic potential generated by a pair of coils in the first type of sub-domain;

[0024] The vector magnetic potential generated by a pair of coils in the first type of sub-domain is:

[0025]

[0026] In Equation (3),, A m1 , B m1 , C m1 , D m1 are the first undetermined coefficients, m is the harmonic order, and μ 0 is the magnetic permeability of air; r is the radius of any point in the polar coordinate system of the first type of sub-domain, θ is the angle from any point to the 0° median line; R s is the inner radius of the stator core, and R r is the outer radius of the rotor core, and i c is the current of the coil in the air gap; lnρ is shown as follows:

[0027]

[0028] In Equation (4), a and ζ are the positions of a pair of coils in the polar coordinate system;

[0029] When r ≥ a, the radial magnetic density in the first type of sub-domain is:

[0030]

[0031] In Equation (5), G 1m =(r / R s ) m +(R r / R s ) m (r / R r ) -m (6)

[0032]

[0033] The tangential magnetic flux density in the first type of subdomain is:

[0034]

[0035] In Equation (8), G 2m =(r / R s ) m -(R r / R s ) m (r / R r ) -m (9)

[0036] The vector magnetic potential generated by the current on the jth permanent magnet pole surface is:

[0037]

[0038] In Equation (10), A mj and C mj are undetermined coefficients, and (j - 1)π / P is the angle between the center line of the first pole and the center line of the jth pole;

[0039] When r < a, the radial magnetic flux density in the first type of subdomain is:

[0040]

[0041] The tangential magnetic flux density in the first type of subdomain is:

[0042]

[0043] In Equation (14),

[0044] The vector magnetic potential generated by the current on the jth permanent magnet pole surface is:

[0045]

[0046] The calculation of the magnetic field in the second type of subdomain includes:

[0047] The Laplace equation in the second type of subdomain is:

[0048]

[0049] In Equation (17), A z2i is the vector magnetic potential of the second type of subdomain;

[0050] The vector magnetic potential of the second type of subdomain is:

[0051]

[0052] In Equation (18), θ iis the angle between the center line of the i-th slot and the 0° line, C l2i and D l2i are undetermined coefficients, R so is the radius of the bottom of the slot opening, E l = lπ / b so , where l is the harmonic order of the magnetic density at the slot opening, b so is the radian of the slot opening width;

[0053] The radial magnetic density of the second type of sub-domain is:

[0054]

[0055] The tangential magnetic density of the second type of sub-domain is:

[0056]

[0057] The calculation of the magnetic field in the third type of sub-domain includes:

[0058] The vector magnetic potential in the third type of sub-domain satisfies the Poisson equation:

[0059]

[0060] In Equation (21), J is the current density in the slot; A z3i is the vector magnetic potential in the third type of sub-domain;

[0061] The current density in the i-th slot of the non-overlapping winding is:

[0062]

[0063] In Equation (22), J i0 = J i1 + J i2 ,

[0064] The vector magnetic potential in the third type of sub-domain is:

[0065]

[0066] In Equation (23), C k3i is an undetermined coefficient, E k = kπ / b sa ;

[0067] The radial magnetic density in the third type of sub-domain is:

[0068]

[0069] The tangential magnetic density in the third type of sub-domain is:

[0070]

[0071] Obtaining the radial magnetic density and tangential magnetic density of a pair of coils in the permanent magnet by combining the magnetic field calculation results of three sub-domains includes:

[0072] Combining the radial magnetic densities of the first type of sub-domain, the second type of sub-domain, and the third type of sub-domain to obtain the radial magnetic density of a pair of coils in the permanent magnet; combining the tangential magnetic densities of the first type of sub-domain, the second type of sub-domain, and the third type of sub-domain to obtain the tangential magnetic density of a pair of coils in the permanent magnet.

[0073] Calculating the magnetic field using the superposition principle to obtain the radial magnetic density and tangential magnetic density throughout the permanent magnet means superimposing the vector magnetic potentials generated by the equivalent surface currents on each side of the surface-mounted permanent magnet to obtain the radial magnetic density and tangential magnetic density in the permanent magnet.

[0074] The vector magnetic potentials generated by the equivalent surface currents on each side of the surface-mounted permanent magnet include the vector magnetic potentials generated by the equivalent surface currents on two straight sides, the vector magnetic potential generated by the equivalent surface current of the long arc, and the vector magnetic potential generated by the equivalent surface current of the short arc; the superposition of the vector magnetic potentials generated by the equivalent surface currents on two straight sides is:

[0075]

[0076] In formula (26), A z1 is the vector magnetic potential generated by a pair of coils in the first type of sub-domain, J 1 is the surface current density of the two straight sides of the permanent magnet, Δr is the length element of the two straight sides of the permanent magnet, k 1 ={1, 2, …, h max / Δr} (27);

[0077] The superposition of the vector magnetic potential generated by the equivalent surface current of the long arc is:

[0078]

[0079] In formula (28), Δγ 1 is the radian element of the long arc of the permanent magnet, the center of the circle is at point O1, J 2 is the surface current density of the long arc of the permanent magnet, R 2 is the radius of the long arc of the permanent magnet, k 2 ={1, 2, …, ζ′ max}, OB is the distance from the center of the motor to the end point B of the long arc of the permanent magnet, OE is the distance from the center of the motor to any point E on the long arc of the permanent magnet, α p is the pole arc coefficient of the permanent magnet, n p is the number of pole pairs of the permanent magnet;

[0080] The superposition of the vector magnetic potential generated by the equivalent surface current of the short arc is:

[0081]

[0082] In Equation (29), A z1 is the vector magnetic potential generated by a pair of coils in the first type of sub-domain, and Δγ 2 is the radian micro-element of the short arc of the permanent magnet, with the center of the circle at point O, and k 3 = {1, 2, …, [α p π / (2n P )] / Δγ 2}(30), and J 3 is the surface current density of the short arc of the permanent magnet.

[0083] The said S2 includes the following steps:

[0084] S21. Represent the radial magnetic density in the first type of sub-domain in the reference coordinate system;

[0085] The radial magnetic density in the first type of sub-domain in the rotor reference coordinate system is:

[0086]

[0087] In Equation (31), A m1 and C m1 are undetermined coefficients related to the permanent magnet excitation, slotting boundary, and armature current. θ r is the rotor position angle, ω r is the rotor rotational angular velocity, and t is time;

[0088] S22. Derive the expression of the contribution of the spatio-temporal variation of the radial magnetic density generated by a pair of equivalent surface currents of the permanent magnet to the rotor eddy current in combination with the relationship between the spatio-temporal variation of the radial magnetic density and the contribution of the rotor eddy current; and extend the expression to the entire permanent magnet;

[0089] It is only related to the excitation of the permanent magnet and has nothing to do with the slotting boundary and armature current. This term does not contribute to the eddy currents of the permanent magnet and the rotor sheath. Therefore, the expression of the contribution of the spatio-temporal variation of the radial magnetic density generated by a pair of equivalent surface currents of the permanent magnet to the rotor eddy current is:

[0090]

[0091] After extending the expression to the entire permanent magnet, it is:

[0092]

[0093] In Equation (33), when calculating the magnetic field generated by the armature current, the armature current only needs to be calculated once. A mk and C mkis the space-time coefficient to be determined for the magnetic field generated by the equivalent current of the k-th permanent magnet varying with time and rotor position; the A mk and C mk are expressed by Fourier series as:

[0094]

[0095] In Equation (34), p is the number of pole pairs, n = 1, 2, 3,..., represents different rotor positions, A mkn and C mkn are the amplitudes of the Fourier series, α n and β n are the phase angles of the Fourier series;

[0096] S23. Express the radial magnetic density generated by the permanent magnet and the armature current using space-time Fourier series, and expand it into an odd Fourier series along the axial length of the motor;

[0097] The expression of the radial magnetic density generated by the permanent magnet and the armature current using space-time Fourier series means substituting Equation (34) into Equation (33), and using trigonometric transformation to obtain the radial magnetic density generated by the permanent magnet and the armature current expressed by space-time Fourier series:

[0098]

[0099] In Equation (35),

[0100] The expansion into an odd Fourier series along the axial length of the motor means that in three-dimensional space, the radial magnetic density generated by the permanent magnet and the armature current is expanded along the axial length and expressed by an odd Fourier series as:

[0101]

[0102] S24. Obtain the axial component and tangential component of the three-dimensional eddy current density through the time derivative of the magnetic density in the conductor, Faraday's law, and the current vector;

[0103] The time derivative of the magnetic density in the conductor is:

[0104]

[0105] In Equation (37),

[0106]

[0107] The divergence of the eddy current density is zero, so the current vector satisfies The edge current within the permanent magnet or the rotor sheath is zero, so the boundary current vector must satisfy According to Faraday's law, we get:

[0108]

[0109] In Equation (38), σ is the magnetic permeability of the conductor;

[0110] Then the radial component of the current vector is:

[0111]

[0112] Equation (39) satisfies the boundary conditions:

[0113]

[0114] In Equation (40), v is the axial harmonic order of the magnetic flux density of the permanent magnet motor, and l eff is the effective axial length of the permanent magnet motor,

[0115] Substituting Equation (37) into Equation (39), the axial component of the three-dimensional eddy current density is:

[0116]

[0117] The tangential component of the three-dimensional eddy current density is:

[0118]

[0119] The analytical calculation model of the three-dimensional eddy current loss is:

[0120]

[0121] In Equation (43), n z 、n θ and n r are respectively the number of axial segments, the number of tangential segments, and the number of radial segments of the rotor conductor.

[0122] Compared with the prior art, the beneficial technical effects brought by the present invention are as follows:

[0123] 1. The present invention proposes an accurate analytical calculation method for the three-dimensional eddy current loss of the surface-mounted permanent magnet motor rotor, aiming to deeply understand the mechanism of rotor eddy current loss generation and optimize and reduce the three-dimensional eddy current loss of the permanent magnet motor rotor, so that the permanent magnet motor operates within a safe range and improves its electromagnetic and thermal performance.

[0124] 2. The three-dimensional finite element calculation of eddy current loss in permanent magnet motors requires high computer configuration (such as high-performance workstations) and long calculation time. The analytical calculation method of permanent magnet motors can usually reduce the computer configuration and save calculation time. Therefore, the present invention can quickly and accurately calculate the three-dimensional eddy current loss of the rotor of surface-mounted permanent magnet motors under the condition of ordinary computer configuration.

[0125] 3. The present invention fully considers the boundary condition that the eddy current density at the end of the permanent magnet or the rotor sheath is 0, ensuring the accuracy of calculating the three-dimensional eddy current loss of the rotor of surface-mounted permanent magnet motors. Brief Description of the Drawings

[0126] Figure 1 It is a schematic diagram of the equivalent surface current of the permanent magnet (parallel magnetization);

[0127] Figure 2 It is a schematic diagram of the equivalent surface current of the permanent magnet (radial magnetization);

[0128] Figure 3 It is a sub-domain model of a surface-mounted high-speed permanent magnet motor with a pair of coils in the air gap and armature current in the slots;

[0129] Figure 4 It is the magnetic flux density in the sheath when the permanent magnet motor is loaded;

[0130] Figure 5 It is the magnetic flux density in the permanent magnet when the permanent magnet motor is loaded. Detailed Embodiments

[0131] Next, the technical solutions of the present invention will be clearly and completely described in conjunction with specific embodiments. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

[0132] Embodiment 1

[0133] As another preferred embodiment of the present invention, this embodiment discloses an analytical calculation method for the three-dimensional eddy current loss of the rotor of a surface-mounted permanent magnet motor, including the following steps:

[0134] S1. Magnetic field analysis of the surface-mounted permanent magnet motor; the magnetic field analysis in this embodiment adopts the prior art, such as the prior art with the publication number of CN116362148A;

[0135] S2. Expand the magnetic field obtained by the analytical calculation to the axial length, and calculate the axial and tangential components of the three-dimensional eddy current density of the rotor;

[0136] S3. Based on the axial and tangential components of the three-dimensional eddy current density, combined with the integral form for calculating the resistance loss after passing current through the resistor, establish an analytical calculation model for three-dimensional eddy current loss;

[0137] S4. Obtain the three-dimensional eddy current loss of the surface-mounted permanent magnet motor rotor according to the analytical calculation model for three-dimensional eddy current loss established in S3.

[0138] In this embodiment, a precise analytical calculation method for the three-dimensional eddy current loss of the surface-mounted permanent magnet motor rotor is proposed, aiming to deeply understand the mechanism of rotor eddy current loss generation, optimize and reduce the three-dimensional eddy current loss of the permanent magnet motor rotor, enable the permanent magnet motor to operate within a safe range, and improve its electromagnetic and thermal performance.

[0139] Embodiment 2

[0140] As another preferred embodiment of the present invention, this embodiment discloses an analytical calculation method for the three-dimensional eddy current loss of the surface-mounted permanent magnet motor rotor, including the following steps:

[0141] S1. Analyze the magnetic field of the surface-mounted permanent magnet motor;

[0142] Obtain the equivalent surface current of the permanent magnet;

[0143] Parallel magnetization and radial magnetization are two main magnetization methods of surface-mounted high-speed permanent magnet motors. The equivalent surface currents of permanent magnets with different thicknesses are as Figure 1 , 2 shown. Figure 1 represents the parallel magnetization of the permanent magnet, Figure 2 represents the radial magnetization of the permanent magnet. The surface current is distributed around the parallel-magnetized permanent magnet, but the surface current is not distributed on the AD arc surface of the radially magnetized permanent magnet;

[0144] The equivalent surface current of the permanent magnet is obtained by the following formula:

[0145] J PM = H cj sinα (1)

[0146] In formula (1), H cj is the coercivity of the permanent magnet, and α is the angle between the magnetization direction and the normal of the permanent magnet surface.

[0147] Establish a sub-domain model of the surface-mounted high-speed permanent magnet motor with a pair of coils in the air gap and armature current in the slot, as Figure 3 ; α and β represent the two conductors of a pair of coils in the air gap. a and ζ represent the positions of a pair of coils in the polar coordinate system. i c is the current of the coil in the air gap, R r is the outer radius of the rotor core, R s is the inner radius of the stator core, R so is the bottom radius of the slot opening, Rsb is the radius at the bottom of the slot, R s1 is the outer radius of the stator, b so is the radian of the slot opening width, b sa is the radian of the slot width, J i1 and J i2 are the current densities of the two-layer windings in the i-th slot. The entire magnetic field can be divided into three simple sub-domains. The first type of sub-domain is the region i between the rotor core and the inner circle of the stator core. The second type of sub-domain is the region 2i in the stator slot opening. The third type of sub-domain is the region 3i in the stator slot. The sub-domain boundaries and the intersection lines of the sub-domains are all parallel to the two coordinate axes of the polar coordinate system.

[0148] Calculate the magnetic field of its corresponding sub-domain, and then combine the magnetic field calculation results of the three sub-domains to obtain the radial magnetic density and tangential magnetic density of a pair of coils in the permanent magnet;

[0149] Among them, the magnetic field of the first type of sub-domain:

[0150] The vector magnetic potential in the first type of sub-domain only has a Z-axis component and satisfies the Laplace equation:

[0151]

[0152] In Equation 2, r is the radius of any point in the polar coordinate system of the first type of sub-domain, θ is the angle of any point to the 0° median line, and A z1 is the vector magnetic potential generated by a pair of coils in the first type of sub-domain;

[0153] The vector magnetic potential generated by a pair of coils in the first type of sub-domain is:

[0154]

[0155] In Equation (3), A m1 , B m1 , C m1 , D m1 are the first undetermined coefficients, m is the harmonic order, and μ 0 is the magnetic permeability of air; r is the radius of any point in the polar coordinate system of the first type of sub-domain, θ is the angle of any point to the 0° median line; R s is the inner radius of the stator core, R r is the outer radius of the rotor core, i c is the current of the coil in the air gap; ln ρ is shown as follows:

[0156]

[0157] In Equation (4), a and ζ are the positions of a pair of coils in the polar coordinate system;

[0158] When r ≥ a, the radial magnetic density in the first type of sub-domain is:

[0159]

[0160] In Equation (5), G 1m =(r / R s ) m +(R r / R s ) m (r / R r ) -m (6)

[0161]

[0162] The tangential magnetic flux density in the first type of sub-domain is:

[0163]

[0164] In Equation (8), G 2m =(r / R s ) m -(R r / R s ) m (r / R r ) -m (9),

[0165] The vector magnetic potential generated by the current on the j-th permanent magnet pole surface is:

[0166]

[0167] In Equation (10), A mj and C mj are undetermined coefficients, and (j - 1)π / P is the angle between the center line of the first pole and the center line of the j-th pole;

[0168] When r < a, the vector magnetic potential in the first type of sub-domain is

[0169] A z1 =∑ m A m1 G 1m cos(mθ)+∑ m (C m1 G 1m +G 3m )sin(mθ) (11)

[0170] In Equation (11),

[0171] When r < a, the radial magnetic flux density in the first type of sub-domain is:

[0172]

[0173] The tangential magnetic flux density in the first type of sub-domain is:

[0174]

[0175] In Equation (14),

[0176] The vector magnetic potential generated by the j-th permanent magnet pole surface current is:

[0177]

[0178]

[0179] The magnetic field in the second type of sub-domain:

[0180] The Laplace equation in the second type of sub-domain is:

[0181]

[0182] In Equation (17), A z2i is the vector magnetic potential of the second type of sub-domain;

[0183] The vector magnetic potential of the second type of sub-domain is:

[0184]

[0185] In Equation (18), θ i is the angle between the center line of the i-th slot and the 0° line, C l2i and D l2i are undetermined coefficients, R so is the radius at the bottom of the slot opening, E l = lπ / b so , where l is the harmonic order of the magnetic flux density at the slot opening, b so is the radian of the slot opening width;

[0186] The radial magnetic flux density in the second type of sub-domain is:

[0187]

[0188] The tangential magnetic flux density in the second type of sub-domain is:

[0189]

[0190] The magnetic field in the third type of sub-domain:

[0191] The vector magnetic potential in the third type of sub-domain satisfies the Poisson equation:

[0192]

[0193] In Equation (21), J is the current density in the slot; A z3iis the vector magnetic potential in the third type of sub-domain;

[0194] The current density of the non-overlapping winding in the i-th slot is:

[0195]

[0196] In Equation (22), J i0 = J i1 + J i2 ,

[0197] The vector magnetic potential in the third type of sub-domain is:

[0198]

[0199] In Equation (23), C k3i is a coefficient to be determined, E k = kπ / b sa ;

[0200] The radial magnetic flux density in the third type of sub-domain is:

[0201]

[0202] The tangential magnetic flux density in the third type of sub-domain is:

[0203]

[0204] By simultaneously solving Equations (5), (13), (19), and (24), the radial magnetic flux density of a pair of coils in the permanent magnet can be calculated. By simultaneously solving Equations (8), (14), (20), and (25), the tangential magnetic flux density of a pair of coils in the permanent magnet can be calculated.

[0205] Apply the superposition principle to calculate the magnetic field and obtain the radial and tangential magnetic flux densities throughout the permanent magnet;

[0206] The superposition principle can be applied to the vector magnetic potential generated by the equivalent surface current of the surface-mounted permanent magnet. When applying the superposition principle, only the vector magnetic potential generated by the current in the primary slot needs to be considered;

[0207] The superposition of the vector magnetic potential generated by the equivalent surface current on the two straight edges is:

[0208]

[0209] In Equation (26), A z1 is the vector magnetic potential generated by a pair of coils in the first type of sub-domain, J 1 is the surface current density of the two straight edges of the permanent magnet, Δr is the length element of the two straight edges of the permanent magnet, k 1 = {1, 2,..., h max / Δr}(27);

[0210] The superposition of the vector magnetic potential generated by the long-arc equivalent surface current is:

[0211]

[0212] In Equation (28), Δγ 1 is the radian element of the long arc of the permanent magnet, with the center at point O1, J 2 is the surface current density of the long arc of the permanent magnet, R 2 is the radius of the long arc of the permanent magnet, k 2 = {1, 2, …, ζ′ max}}, OB is the distance from the center of the motor to the end point B of the long arc of the permanent magnet, OE is the distance from the center of the motor to any point E on the long arc of the permanent magnet, α p is the pole arc coefficient of the permanent magnet, n p is the number of pole pairs of the permanent magnet;

[0213] The superposition of the vector magnetic potential generated by the short-arc equivalent surface current is:

[0214]

[0215] In Equation (29), Δγ 2 is the radian element of the short arc of the permanent magnet, with the center at point O, and k 3 = {1, 2, …, [α p π / (2n P )] / Δγ 2}(30), J 3 is the surface current density of the short arc of the permanent magnet.

[0216] According to Equations (26), (28), and (29), after the superposition of the vector magnetic potential generated by the equivalent surface current on each side of the permanent magnet, the radial magnetic density and tangential magnetic density inside the permanent magnet can be obtained.

[0217] S2. Extend the magnetic field obtained by the analytical calculation to the axial length, and calculate the axial component and tangential component of the three-dimensional eddy current density of the rotor;

[0218] According to Equation (5), the magnetic density of a pair of coils inside the permanent magnet in the rotor coordinate system at θ = θ r + ω r t is expressed as:

[0219]

[0220] In Equation (31), A m1 and C m1 are undetermined coefficients related to the permanent magnet excitation, slotting boundary, and armature current, θ ris the rotor position angle, ω r is the rotor angular velocity of rotation, and t is time;

[0221] is only related to the excitation of the permanent magnet and has nothing to do with the slot boundary and armature current. This term makes no contribution to the eddy current of the permanent magnet and the rotor sheath. Therefore, the expression for the contribution of the spatio-temporal variation of the radial magnetic density generated by a pair of equivalent surface currents of the permanent magnet to the rotor eddy current is:

[0222]

[0223] When extended to the entire permanent magnet, the above formula can be expressed as:

[0224]

[0225] In Equation (33), when calculating the magnetic field generated by the armature current, the armature current only needs to be calculated once, A mk and C mk are the undetermined spatio-temporal coefficients of the magnetic field generated by the equivalent current of the k-th permanent magnet varying with time and rotor position; the A mk and C mk are expressed by Fourier series as:

[0226]

[0227] In Equation (34), p is the number of pole pairs, n = 1, 2, 3,..., representing different rotor positions, A mkn and C mkn are the amplitudes of the Fourier series, and α n and β n are the phase angles of the Fourier series;

[0228] Substitute Equation (34) into Equation (33), and use trigonometric transformation to obtain the radial magnetic density generated by the permanent magnet and the armature current expressed by the spatio-temporal Fourier series:

[0229]

[0230] In Equation (35),

[0231] In three-dimensional space, the radial magnetic density B erp (r, θ, t) can be expanded into an odd Fourier series along the axial length of the motor:

[0232]

[0233] Equation (36) can ensure that the eddy current at the ends of the permanent magnet and the rotor sheath is zero.

[0234] The time derivative of the magnetic flux density in a conductor can be used to calculate eddy currents, and the time derivative of the magnetic flux density in the conductor is:

[0235]

[0236]

[0237] In Equation (37),

[0238]

[0239] The divergence of the eddy current density is zero, so the current vector satisfies The edge current in the permanent magnet or the rotor sheath is zero, so the boundary current vector must satisfy According to Faraday's law, we get:

[0240]

[0241] In Equation (38), σ is the magnetic permeability of the conductor;

[0242] Then the radial component of the current vector is:

[0243]

[0244] Equation (39) satisfies the boundary conditions:

[0245]

[0246] In Equation (40), v is the axial harmonic order of the magnetic flux density of the permanent magnet motor, and l eff is the axial effective length of the permanent magnet motor,

[0247] Substituting Equation (37) into Equation (39), the axial component of the three-dimensional eddy current density is:

[0248]

[0249] The tangential component of the three-dimensional eddy current density is:

[0250]

[0251] S3. Based on the axial and tangential components of the three-dimensional eddy current density, a three-dimensional eddy current loss analytical calculation model is established as:

[0252]

[0253] In Equation (43), n z 、n θ and nr They are respectively the axial segmentation number, tangential segmentation number, and radial segmentation of the rotor conductor.

[0254] S4. Obtain the three-dimensional eddy current loss of the surface-mounted permanent magnet motor rotor according to the three-dimensional eddy current loss analytical calculation model established in S3.

[0255] Analysis and calculation case of three-dimensional eddy current loss of surface-mounted high-speed permanent magnet motor rotor

[0256] The case prototype is a permanent magnet motor of 75kW / 33000rpm, and the parameters are shown in Table 1.

[0257] Table 1 Motor performance parameters

[0258] Performance parameter Value Performance parameter Value Rated power 75 Outer diameter of stator 152 Rated voltage 380 Inner diameter of stator 63.5 Rated current 120 Outer diameter of sheath 59.3 Rated efficiency 96.9％ Outer diameter of permanent magnet 57.9 Rated power factor 0.98 Inner diameter of permanent magnet 46.5 Rated speed 33000 Slot opening width (mm) 2.5 Rotor sheath material Titanium alloy Tc4 Slot opening height (mm) 1 Permanent magnet material Samarium cobalt Number of turns per slot 8 Core length 140 Number of parallel branches 1 Chording factor 5 / 6

[0259] The peak value of the load current of the permanent magnet motor is 151A. Figure 4 / 5 are the magnetic fluxes in the Tc4 sheath (r = 29.3mm) and the permanent magnet (r = 28mm) of the permanent magnet motor under load. The three-dimensional eddy current loss of the rotor calculated by the method of the present invention is 215W, while the eddy current loss of the rotor obtained by three-dimensional finite element calculation is 224W, and the error is small. Therefore, the result obtained by the analytical calculation method of the three-dimensional eddy current loss of the surface-mounted high-speed permanent magnet motor rotor of the present invention has high accuracy.

Claims

1. A method for calculating the three-dimensional eddy current loss of a surface-mounted permanent magnet motor rotor, characterized in that: The method includes the following steps: S1. Analytical solution of the magnetic field of the surface-mounted permanent magnet motor; The analytical solution of the magnetic field of the surface-mounted permanent magnet motor includes: Obtaining the equivalent surface current of the permanent magnet; Establishing a sub-domain model of the surface-mounted high-speed permanent magnet motor for a pair of coils in the air gap and the armature current in the slots; dividing it into three sub-domains, calculating the magnetic fields of their corresponding sub-domains, and then combining the calculation results of the magnetic fields of the three sub-domains to obtain the radial magnetic density and tangential magnetic density of a pair of coils in the permanent magnet; Calculating the magnetic field by using the superposition principle to obtain the radial magnetic density and tangential magnetic density within the entire permanent magnet; S2. Extending the magnetic field obtained by analytical calculation to the axial length and calculating the axial component and tangential component of the three-dimensional eddy current density of the rotor; S3. According to the axial component and tangential component of the three-dimensional eddy current density, combined with the integral form of calculating the resistance loss after passing current through the resistor, establishing an analytical calculation model for the three-dimensional eddy current loss; The analytical calculation model for the three-dimensional eddy current loss is: In formula (43), n z 、n θ and n r are the axial segment number, tangential segment number and radial segment number of the rotor conductor respectively; S4. Obtaining the three-dimensional eddy current loss of the rotor of the surface-mounted permanent magnet motor according to the analytical calculation model for the three-dimensional eddy current loss established in S3.

2. The method for calculating the three-dimensional eddy current loss of a surface-mounted permanent magnet motor rotor according to claim 1, characterized in that: The equivalent surface current of the permanent magnet is obtained by the following formula: J PM =H cj sinα (1) In formula (1), H cj is the coercive force of the permanent magnet, and α is the angle between the magnetization direction and the normal direction of the permanent magnet surface.

3. The method for calculating the three-dimensional eddy current loss of a surface-mounted permanent magnet motor rotor according to claim 1, characterized in that: The three sub-domains include the first type of sub-domain i between the rotor core and the inner circle of the stator core, the second type of sub-domain 2i in the stator slot opening, and the third type of sub-domain 3i in the stator slot.

4. The method for calculating the three-dimensional eddy current loss of a surface-mounted permanent magnet motor rotor according to claim 3, characterized in that: The calculation of the magnetic field of the first type of sub-domain includes: There is only a Z-axis component in the first type of sub-domain, and the vector magnetic potential that satisfies the Laplace equation: In formula (2), r is the radius of any point in the polar coordinate system of the first subdomain, θ is the angle from any point to the 0° midline, and A z1 is the vector magnetic potential generated by a pair of coils in the first type of subdomain; The vector magnetic potential generated by a pair of coils in the first type of sub-domain is: In formula (3), A m1 ,B m1 , C m1 , D m1 is the first undetermined coefficient, m is the harmonic order, μ0 is the magnetic permeability of air; r is the radius of any point in the polar coordinate system of the first subdomain, θ is the angle from any point to the 0° midline; R s is the inner radius of the stator core, R r is the outer radius of the rotor core, i c is the current of the coil in the air gap; lnρ is shown as follows: In formula (4), a and ζ are the positions of a pair of coils in the polar coordinate system; When r≥a, the radial magnetic density in the first type of sub-domain is: In formula (5), G 1m =(r / R s ) m +(R r / R s ) m (r / R r ) -m (6) The tangential magnetic density in the first type of sub-domain is: In formula (8), G 2m =(r / R s ) m -(R r / R s ) m (r / R r ) -m (9) The vector magnetic potential generated by the surface current of the jth permanent magnet pole face is: In formula (10), A mj and C mj is the unknown coefficient, (j-1)π / P is the angle between the center line of the first magnetic pole and the center line of the jth magnetic pole; When r<a, the radial magnetic density in the first type of sub-domain is: The tangential magnetic density in the first type of sub-domain is: In formula (14), The vector magnetic potential generated by the surface current of the jth permanent magnet pole face is: The calculation of the magnetic field of the second type of sub-domain includes: The Laplace equation in the second type of sub-domain is: In formula (17), A z2i is the vector magnetic potential of the second type of subdomain; The vector magnetic potential of the second type of sub-domain is: In formula (18), θ i is the angle between the center line of the i-th slot and the 0° line, C l2i and D l2i is the unknown coefficient, R so is the radius of the bottom of the slot, E l =lπ / b so , where l is the harmonic order of the slot flux density, b so is the arc of the slot width; The radial magnetic density of the second type of sub-domain is: The tangential magnetic density of the second type of sub-domain is: The calculation of the magnetic field of the third type of sub-domain includes: The vector magnetic potential of the third type of sub-domain satisfies the Poisson equation: In formula (21), J is the current density in the cell; A z3i is the vector magnetic potential in the third subdomain; The current density of the non-overlapping winding in the ith slot is: In formula (22), J i0 =J i1 +J i2 , The vector magnetic potential in the third type of sub-domain is: In formula (23), C k3i is the coefficient to be determined, The radial magnetic density in the third type of sub-domain is: The tangential magnetic density in the third type of sub-domain is:

5. The method for calculating the three-dimensional eddy current loss of a surface-mounted permanent magnet motor rotor according to claim 4, characterized in that: The combination of the calculation results of the magnetic fields of the three sub-domains to obtain the radial magnetic density and tangential magnetic density of a pair of coils in the permanent magnet includes: Combining the radial magnetic densities of the first type of sub-domain, the second type of sub-domain, and the third type of sub-domain to obtain the radial magnetic density of a pair of coils in the permanent magnet; combining the tangential magnetic densities of the first type of sub-domain, the second type of sub-domain, and the third type of sub-domain to obtain the tangential magnetic density of a pair of coils in the permanent magnet.

6. A method for calculating the three-dimensional eddy current loss of a surface-mounted permanent magnet motor rotor according to claim 5, characterized in that: The calculation of the magnetic field by using the superposition principle to obtain the radial magnetic density and tangential magnetic density within the entire permanent magnet means that the vector magnetic potentials generated by the equivalent surface currents of each side of the surface-mounted permanent magnet are superimposed to obtain the radial magnetic density and tangential magnetic density within the permanent magnet.

7. The method for calculating the three-dimensional eddy current loss of a surface-mounted permanent magnet motor rotor according to claim 6, characterized in that: The vector magnetic potential generated by the equivalent surface current of each side of the surface-mounted permanent magnet includes the vector magnetic potential generated by the equivalent surface current of the two straight edges, the vector magnetic potential generated by the long arc equivalent surface current and the vector magnetic potential generated by the short arc equivalent surface current; the vector magnetic potential generated by the equivalent surface current of the two straight edges is superimposed as: In formula (26), A z1 is the vector magnetic potential generated by a pair of coils in the first subdomain, J1 is the surface current density of the two straight sides of the permanent magnet, Δr is the length of the two straight sides of the permanent magnet, k1={1,2,…,h max / Δr}(27); The vector magnetic potential superposition generated by the long arc equivalent surface current is: In formula (28), Δγ1 is the arc microelement of the permanent magnet long arc, the center of the circle is at point O1, J2 is the surface current density of the permanent magnet long arc, R2 is the radius of the permanent magnet long arc, k2 = {1,2,…,ζ′ max }, OB is the distance from the center of the motor to the end point B of the permanent magnet long arc, OE is the distance from the center of the motor to any point E on the permanent magnet long arc, α p is the permanent magnet pole arc coefficient, n p is the number of permanent magnet pole pairs; The vector magnetic potential superposition generated by the short arc equivalent surface current is: In formula (29), A z1 is the vector magnetic potential generated by a pair of coils in the first subdomain, Δγ2 is the arc element of the permanent magnet short arc, the center of the circle is at point O, and k3 = {1,2,…,[α p π / (2n P )] / Δγ2} (30), J3 is the surface current density of the permanent magnet short arc.

8. A method for analyzing and calculating three-dimensional eddy current loss of a surface-mounted permanent magnet motor rotor as claimed in claim 1 or 7, characterized in that: The S2 comprises the following steps: S21, expressing the radial magnetic flux density in the first subdomain in a reference coordinate system; The radial magnetic flux density in the first sub-domain in the rotor reference coordinate system is: In formula (31), A m1 and C m1 are the unknown coefficients related to permanent magnet excitation, slot boundaries, and armature current, θ r is the rotor position angle, ω r is the rotor rotation angular velocity, t is the time; S22. Based on the relationship between the spatial and temporal variation of radial magnetic flux and the contribution of rotor eddy current, derive the expression for the contribution of spatial and temporal variation of radial magnetic flux to rotor eddy current generated by a pair of equivalent surface currents of permanent magnets; and extend the expression to the entire permanent magnet; It is only related to the excitation of the permanent magnet, and has nothing to do with the slot boundary and the armature current. This item does not contribute to the eddy current of the permanent magnet and the rotor sleeve. Therefore, the expression for the contribution of the spatiotemporal variation of the radial flux density generated by a pair of equivalent surface currents of the permanent magnet to the rotor eddy current is: After expanding the expression to the entire permanent magnet, it becomes: In formula (33), when calculating the magnetic field generated by the armature current, the armature current only needs to be calculated once, A mk and C mk is the time-space coefficient of the kth permanent magnet equivalent current generating magnetic field that changes with time and rotor position; the A mk and C mk Expressed in Fourier series: In formula (34), p is the number of pole pairs, n = 1, 2, 3, ..., representing different rotor positions, A mkn and C mkn is the amplitude of the Fourier series, α n and β n is the phase angle of the Fourier series; S23, expressing the radial magnetic flux density generated by the permanent magnet and the armature current by using a time-space Fourier series, and expanding it into an odd Fourier series along the axial length of the motor; The radial magnetic flux generated by the permanent magnet and the armature current is expressed by the space-time Fourier series. This means that equation (34) is substituted into equation (33) and trigonometric function transformation is used to obtain the radial magnetic flux generated by the permanent magnet and the armature current expressed by the space-time Fourier series: In formula (35), The expansion into an odd Fourier series in the axial length of the motor means that in three-dimensional space, the radial magnetic flux density is expanded in the axial length and expressed in an odd Fourier series as: S24, obtaining the axial component and tangential component of the three-dimensional eddy current density through the differential of the magnetic flux density with respect to time in the conductor, Faraday's law and the current vector; The differential of the magnetic flux density in the conductor with respect to time is: In formula (37), Eddy current density The divergence of is zero, so the current vector satisfies The edge current in the permanent magnet or rotor casing is zero, so the boundary current vector must satisfy According to Faraday's law, we get: In formula (38), σ is the magnetic permeability of the conductor; Then the radial component of the current vector is: Formula (39) satisfies the boundary conditions: In formula (40), ν is the axial harmonic order of the permanent magnet motor magnetic flux density, l eff is the effective axial length of the permanent magnet motor, Substituting equation (37) into equation (39), we can obtain the axial component of the three-dimensional eddy current density: The tangential component of the three-dimensional eddy current density is:

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