Modeling method of permanent magnet synchronous motor based on stator and rotor eigen characteristics

Abstract

This invention belongs to the field of permanent magnet synchronous motor control, specifically involving a modeling method for permanent magnet synchronous motors based on the intrinsic characteristics of the stator and rotor. It aims to solve the problems of difficulty in accurately obtaining key parameters and the high difficulty and poor representativeness of linear and nonlinear analytical models of the motor in existing permanent magnet synchronous motor modeling. This invention includes: using the winding function N... a,b,c (γ), inverted air gap function g ‑1 (γ,θ) and the empty air gap magnetic flux density B gap The three intrinsic characteristics (γ, θ) are used to accurately describe the permanent magnet synchronous motor (PMSM). Taking into account the non-sinusoidal distribution of the stator windings and the influence of non-ideal factors such as rotor magnetic field harmonics, linear and nonlinear analytical models of the PMSM are constructed. This invention establishes the correspondence between the motor structure, material properties, and other factors and the analytical model of the motor. Using known intrinsic parameters of the motor, the relevant parameters in the established analytical model can be solved, effectively solving the problem of accurately obtaining model parameters and improving the accuracy of the PMSM model.

Description

Technical Field

[0001] This invention belongs to the field of permanent magnet synchronous motor control, and specifically relates to a modeling method for permanent magnet synchronous motors based on the intrinsic characteristics of stator and rotor. Background Technology

[0002] AC permanent magnet synchronous motors (PMSMs) are characterized by high power density, long service life, and excellent speed regulation performance, making them widely used in new energy vehicle drives and flywheel energy storage. For many years, the operating theory of PMSMs has been primarily based on fundamental wave analysis theory using time phasor and space vector analysis. This type of model is only suitable for analyzing physical quantities characterizing the fundamental wave operating features of the motor, such as average electromagnetic torque, fundamental voltage, and fundamental current, or for designing control laws in the fundamental wave plane. When studying torque ripple, vibration noise, or performing voltage and current harmonic analysis of PMSMs, it is necessary to consider more complex nonlinear models that can more accurately describe the actual motor. In particular, if a nonlinear analytical model of the PMSM can be established, analytical solutions for the amplitudes of each harmonic of torque ripple, vibration noise, or voltage and current can be obtained. Using the obtained analytical expressions for each harmonic of the relevant parameters, the related problems caused by motor nonlinearity can be studied more intuitively.

[0003] To establish analytical models of permanent magnet synchronous motors (PMSMs), a common approach is to assume ideal modeling conditions, such as ignoring rotor magnetic field harmonics and assuming symmetrical and sinusoidal stator three-phase windings. This simplifies the complex nonlinear mathematical model into simpler mathematical equations. While this method allows for rapid modeling and verification of PMSMs, the aforementioned assumptions are overly idealistic, preventing the established model from accurately reflecting the nonlinear characteristics of the stator and rotor. As a technological improvement, patent CN201711322621.1, "A Design Method for Discrete Simulation Model of Permanent Magnet Motor," proposes a method to construct an equivalent mathematical model of the motor using the phase voltages va, vb, and vc of the three-phase windings. This method simplifies the data processing and makes it easier for practical simulation applications. However, the equivalent voltage model obtained by this method contains a 3×3 inductance matrix. Each element in this matrix is ​​determined by the intrinsic parameters of the motor stator and rotor, such as the properties of ferromagnetic materials and the winding arrangement. The proposed technical solution does not provide a method for obtaining each element of the inductance matrix. The ease of obtaining each element in the inductance matrix and the accuracy of obtaining the inductance matrix will affect the accuracy of the established model. This is a problem that must be solved when applying the model in practice.

[0004] In nonlinear modeling of permanent magnet synchronous motors (PMSMs), the finite element method (FEM) or magnetic network method are two of the most commonly used methods. Finite element analysis of the magnetic field can accurately obtain parameters such as the motor's inductance and back electromotive force. Although various physical parameters of the motor can be calculated precisely, the calculation of these parameters is very time-consuming. The nonlinearity of parameters such as inductance increases the difficulty of motor modeling. Patent CN201910920662.3, "A Modeling and Electromagnetic Performance Calculation Method for Permanent Magnet Motors Based on Magnetic Networks," introduces the mesh generation concept from the finite element method into the equivalent magnetic circuit method. It generates a corresponding element reluctance model for each meshed element, connects them into a network, and applies boundary and motion conditions. The calculation speed is improved compared to the traditional finite element method, but it is essentially a technical improvement of the finite element method. Like the finite element method, it cannot obtain nonlinear analytical expressions for the voltage or current of the PMSM.

[0005] Therefore, how to provide a modeling method for permanent magnet synchronous motors that establishes the correspondence between factors such as motor structure and material properties and the analytical model of the motor, while comprehensively considering the influence of non-ideal factors such as the non-sinusoidal distribution of the stator windings and rotor magnetic field harmonics, and completes the establishment of the analytical model of the permanent magnet synchronous motor; and how to accurately express each parameter in the established analytical model of the motor through the intrinsic parameters of the stator and rotor, so as to achieve convenient and high-precision acquisition of each parameter in the model, are all technical problems that urgently need to be solved in the process of accurate modeling and practical application of permanent magnet synchronous motors. Summary of the Invention

[0006] To address the aforementioned problems in existing technologies, namely the difficulty in accurately obtaining key parameters in the modeling of permanent magnet synchronous motors (PMSMs), and the high difficulty and poor representativeness of linear and nonlinear analytical models of the motors, this invention provides a PMSM modeling method based on the intrinsic characteristics of the stator and rotor. The PMSM modeling method includes:

[0007] Step S10: Construct the stator voltage equation and flux linkage equation of the three-phase permanent magnet synchronous motor in the natural coordinate system based on Faraday's law;

[0008] Step S20: For a permanent magnet synchronous motor with three-phase symmetrical and sinusoidal windings, the fundamental wave expression and the reverse air gap function expression of the winding function are obtained by taking the a-phase winding as the reference point; for a permanent magnet synchronous motor with non-sinusoidal winding distribution, the harmonic expression of the winding function is obtained by taking the a-phase winding as the reference point.

[0009] Step S30: Based on the fundamental expression of the winding function, using the position of the stator a-phase axis as the starting point of the spatial position, solve for the linear stator inductance of the motor; based on the harmonic expression of the winding function, solve for the nonlinear stator inductance of the motor.

[0010] Step S40: Approximate the air gap magnetic flux density curve generated by the rotor magnetic field of the permanent magnet synchronous motor as a rectangular wave, and solve for the no-load air gap magnetic flux density of the permanent magnet synchronous motor.

[0011] Step S50: Based on the fundamental expression of the winding function, the no-load air gap magnetic flux density of the permanent magnet synchronous motor, and the stator voltage equation and flux linkage equation of the three-phase permanent magnet synchronous motor in the natural coordinate system, the fundamental expression of the permanent magnet flux linkage of the motor rotor is obtained by solving; based on the harmonic expression of the winding function, the no-load air gap magnetic flux density of the permanent magnet synchronous motor, and the stator voltage equation and flux linkage equation of the three-phase permanent magnet synchronous motor in the natural coordinate system, the harmonic expression of the permanent magnet flux linkage of the motor rotor is obtained by solving.

[0012] Step S60: Based on the fundamental expression of the winding function, the reverse air gap function expression, the no-load air gap magnetic flux density, and the fundamental expression of the permanent magnet flux linkage of the motor rotor, a linear analytical model of the permanent magnet synchronous motor is constructed; based on the harmonic expression of the winding function, the reverse air gap function expression, the no-load air gap magnetic flux density, and the harmonic expression of the permanent magnet flux linkage of the motor rotor, a nonlinear analytical model of the permanent magnet synchronous motor is constructed.

[0013] In some preferred embodiments, the stator voltage equation and flux linkage equation of the three-phase permanent magnet synchronous motor in the natural coordinate system are expressed as follows:

[0014]

[0015] Λ=L s I-Λ m

[0016] Where E represents the stator induced electromotive force; u = [u a u b u c ] T , representing the stator phase voltage; I = [i a i b i c ] T Λ represents the stator phase current; R represents the resistance of the stator phase winding; Λ = [λ a λ b λ c ] T , representing stator flux linkage; L represents the permanent magnet flux linkage linked to the stator windings; s This represents the stator inductance matrix.

[0017] In some preferred embodiments, the fundamental expression of the winding function and the reverse air gap function are as follows:

[0018]

[0019]

[0020] Where, N t γ represents the number of turns per phase winding; γ represents the distance from any point in space to the axis of the stator phase a winding; α PM and d PM These represent the width and thickness of the permanent magnet, respectively; d air θ represents the air gap height between the stator and rotor core; θ represents the rotor position angle; k = 2, 4, 6, ..., ∞.

[0021] In some preferred embodiments, the harmonic expression of the winding function is:

[0022]

[0023] in, This represents the winding coefficient.

[0024] In some preferred embodiments, the no-load air gap magnetic flux density of the permanent magnet synchronous motor is expressed as follows:

[0025]

[0026] Among them, B δ α PM These represent the amplitude and width of the approximate rectangular wave, respectively. denoted as the amplitude of the kth harmonic of the air gap magnetic flux density of the rotor magnetic field.

[0027] In some preferred embodiments, the fundamental expression of the permanent magnet flux linkage of the motor rotor is:

[0028]

[0029] in, These represent the permanent magnet flux linkages linked to the three-phase stator windings a, b, and c, respectively; λ m1 =rlπB1N t B1 represents the fundamental amplitude of the permanent magnet flux linkage of the phase winding; r represents the fundamental amplitude of the air gap magnetic flux density of the rotor magnetic field; l represents the stator inner diameter; and N represents the stator axial length. t n represents the number of turns per phase winding; p This represents the number of rotor pole pairs of the motor.

[0030] In some preferred embodiments, the harmonic expression of the permanent magnet flux linkage of the motor rotor is:

[0031]

[0032] Where, λ mk =rlπB k N tB is the amplitude of the kth harmonic of the permanent magnet flux linkage linked by the phase winding; k The amplitude of the kth harmonic of the air gap magnetic flux density of the rotor magnetic field is represented by r; the stator inner diameter is represented by l; and the stator axial length is represented by N. t n represents the number of turns per phase winding; p This represents the number of rotor pole pairs of the motor.

[0033] In some preferred embodiments, the linear analytical model of the permanent magnet synchronous motor is expressed as follows:

[0034]

[0035]

[0036]

[0037] Where L0, L1, and L2 represent the self-inductance coefficient, leakage inductance coefficient, and mutual inductance coefficient, respectively; d ,u q i d i q Representing the stator voltage and current in the dq coordinate system respectively; λ d ,λ q L represents the stator flux linkage in the dq coordinate system; d ,L q Represents the dq-axis stator inductance; λ m1 ω represents the fundamental amplitude of the permanent magnet flux linkage fused to each phase winding of the stator. e It represents the electrical angular velocity of the motor rotor.

[0038] In some preferred embodiments, the self-inductance coefficient, leakage inductance coefficient, and mutual inductance coefficients L0, L1, L2 are expressed as follows:

[0039]

[0040] Where μ0 represents the air permeability, l mt r represents the length of each turn of the stator winding of the motor. c h represents the radius of the stator winding conductor; ci Represents the stator winding insulation thickness; b w It represents the total thickness of all windings stacked in the stator slot.

[0041] In some preferred embodiments, the nonlinear analytical model of the permanent magnet synchronous motor is expressed as follows:

[0042]

[0043]

[0044] Among them, udq =[u d u q ] T I dq =[i d i q ] T u d ,u q i d i q Representing the stator voltage and current in the dq coordinate system, respectively; λ m5 The fifth harmonic amplitudes of the permanent magnet flux linkages in each phase winding of the stator; σ2, σ4, σ6, σ 10 N1 and N5 represent the 2nd, 4th, 6th, and 10th order coefficients of the air gap function, respectively; N1 and N5 represent the first and fifth order coefficients of the winding function, respectively.

[0045] The beneficial effects of this invention are:

[0046] (1) The present invention is a modeling method for permanent magnet synchronous motors based on the intrinsic characteristics of stator and rotor. Based on the winding function theory, the analytical expressions of the voltage and flux linkage equations of permanent magnet synchronous motors expressed by the intrinsic characteristics of stator and rotor are derived. The method comprehensively considers the influence of two main nonlinear factors: the non-sinusoidal distribution of the stator winding and the harmonics of the rotor magnetic field. The method completes the construction of the linear analytical model and the nonlinear analytical model of the permanent magnet synchronous motor in analytical form, and realizes the accurate modeling of the permanent magnet synchronous motor.

[0047] (2) The present invention provides a modeling method for permanent magnet synchronous motors based on the intrinsic characteristics of stator and rotor. It provides a method for obtaining relevant parameters of motor model with clear physical meaning and easy operation. Based on the material properties of stator and rotor of permanent magnet synchronous motor, winding arrangement and stator and rotor dimensions, the analytical expression of permanent magnet synchronous motor model is obtained, and the correspondence between the intrinsic parameters of stator and rotor of motor and the parameters involved in motor model is established. This realizes the accurate expression of relevant parameters in the established motor model and effectively solves the problem of accurate acquisition of key parameters in motor model.

[0048] (3) The modeling method of permanent magnet synchronous motor based on the intrinsic characteristics of stator and rotor in this invention has a clear mechanism for parameter solution and strong practicality. The established linear motor model based on three-phase symmetrical sinusoidal distribution windings and the nonlinear motor model considering the influence of nonlinear factors such as non-sinusoidal distribution of stator windings and rotor magnetic field harmonics are both analytical in form. Moreover, all parameters in the obtained motor model can be obtained by using known intrinsic parameters of the motor. The model-related parameters are easy and quick to obtain, realizing high-precision modeling of permanent magnet synchronous motor.

[0049] (4) The modeling method of permanent magnet synchronous motor based on the intrinsic characteristics of stator and rotor of the present invention can be flexibly selected between linear analytical model and nonlinear analytical model according to actual needs when applied to engineering practice. The proposed method has obvious advantages over existing technologies in terms of constructing practical analytical models of permanent magnet synchronous motors and the accuracy of model-related parameter calculation. The proposed modeling idea can also provide useful reference for analyzing related control problems and designing control strategies. Attached Figure Description

[0050] Other features, objects, and advantages of this application will become more apparent from the following detailed description of non-limiting embodiments with reference to the accompanying drawings:

[0051] Figure 1 This is a schematic diagram of the stator and rotor structure of a permanent magnet synchronous motor according to an embodiment of the modeling method for permanent magnet synchronous motors based on the intrinsic characteristics of stator and rotor of the present invention;

[0052] Figure 2 This is an equivalent circuit diagram of a motor according to an embodiment of the permanent magnet synchronous motor modeling method based on the intrinsic characteristics of stator and rotor of the present invention.

[0053] Figure 3 This is a schematic diagram of the motor winding function waveform of an embodiment of the permanent magnet synchronous motor modeling method based on the intrinsic characteristics of stator and rotor of the present invention;

[0054] Figure 4 This is a schematic diagram of the inverted air gap function waveform of a motor according to an embodiment of the permanent magnet synchronous motor modeling method based on the intrinsic characteristics of stator and rotor of the present invention;

[0055] Figure 5 This is a schematic diagram of the no-load air gap magnetic flux density waveform of a motor rotor based on the modeling method of permanent magnet synchronous motor based on the intrinsic characteristics of stator and rotor of the present invention. Detailed Implementation

[0056] The present application will now be described in further detail with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are for illustrative purposes only and are not intended to limit the invention. Furthermore, it should be noted that, for ease of description, only the parts relevant to the invention are shown in the accompanying drawings.

[0057] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. This application will now be described in detail with reference to the accompanying drawings and embodiments.

[0058] This invention provides a modeling method for permanent magnet synchronous motors (PMSMs) based on the intrinsic characteristics of the stator and rotor. This method, considering the non-sinusoidal distribution of the stator windings and the nonlinearity of the rotor magnetic field harmonics, derives analytical expressions for the voltage and flux linkage equations of the PMSM, thus constructing an analytical model of the motor. It establishes correlation expressions between the intrinsic parameters of the stator and rotor, reflecting information such as ferromagnetic material properties, winding arrangement, and dimensions, and the parameters in the motor model. Based on the intrinsic parameters of the motor, it achieves accurate solutions for the parameters of the established motor model. Starting from the stator and rotor materials and structure of the motor body, the proposed PMSM modeling method, while obtaining the analytical expressions for the nonlinear model of the PMSM, elucidates the correspondence between the intrinsic characteristics of the stator and rotor and the parameters in the model. By using known intrinsic parameters of the stator and rotor, it achieves accurate expression of the parameters in the established model, effectively solving the pain point of accurately obtaining model parameters. The proposed method has significant advantages over existing technologies in constructing practical analytical models of PMSMs and in the accuracy of obtaining relevant model parameters.

[0059] The present invention provides a modeling method for a permanent magnet synchronous motor based on the intrinsic characteristics of the stator and rotor, the modeling method comprising:

[0060] Step S10: Construct the stator voltage equation and flux linkage equation of the three-phase permanent magnet synchronous motor in the natural coordinate system based on Faraday's law;

[0061] Step S20: For a permanent magnet synchronous motor with three-phase symmetrical and sinusoidal windings, the fundamental wave expression and the reverse air gap function expression of the winding function are obtained by taking the a-phase winding as the reference point; for a permanent magnet synchronous motor with non-sinusoidal winding distribution, the harmonic expression of the winding function is obtained by taking the a-phase winding as the reference point.

[0062] Step S30: Based on the fundamental expression of the winding function, using the position of the stator a-phase axis as the starting point of the spatial position, solve for the linear stator inductance of the motor; based on the harmonic expression of the winding function, solve for the nonlinear stator inductance of the motor.

[0063] Step S40: Approximate the air gap magnetic flux density curve generated by the rotor magnetic field of the permanent magnet synchronous motor as a rectangular wave, and solve for the no-load air gap magnetic flux density of the permanent magnet synchronous motor.

[0064] Step S50: Based on the fundamental expression of the winding function, the no-load air gap magnetic flux density of the permanent magnet synchronous motor, and the stator voltage equation and flux linkage equation of the three-phase permanent magnet synchronous motor in the natural coordinate system, the fundamental expression of the permanent magnet flux linkage of the motor rotor is obtained by solving; based on the harmonic expression of the winding function, the no-load air gap magnetic flux density of the permanent magnet synchronous motor, and the stator voltage equation and flux linkage equation of the three-phase permanent magnet synchronous motor in the natural coordinate system, the harmonic expression of the permanent magnet flux linkage of the motor rotor is obtained by solving.

[0065] Step S60: Based on the fundamental expression of the winding function, the reverse air gap function expression, the no-load air gap magnetic flux density, and the fundamental expression of the permanent magnet flux linkage of the motor rotor, a linear analytical model of the permanent magnet synchronous motor is constructed; based on the harmonic expression of the winding function, the reverse air gap function expression, the no-load air gap magnetic flux density, and the harmonic expression of the permanent magnet flux linkage of the motor rotor, a nonlinear analytical model of the permanent magnet synchronous motor is constructed.

[0066] Among them, the winding function N a,b,c (γ), inverted air gap function g -1 (γ,θ) and the empty air gap magnetic flux density B gap (γ,θ) are the three intrinsic characteristics of the permanent magnet synchronous motor used in the modeling of this invention.

[0067] To more clearly illustrate the modeling method for permanent magnet synchronous motors based on the intrinsic characteristics of stator and rotor of the present invention, the following detailed description of each step of the linear and nonlinear modeling of the permanent magnet synchronous motor in the embodiments of the present invention is provided in conjunction with the accompanying drawings.

[0068] The linear modeling method for permanent magnet synchronous motors based on the intrinsic characteristics of stator and rotor according to the first embodiment of the present invention is described in detail below:

[0069] like Figure 1 The diagram shown is a schematic representation of the stator and rotor structure of a permanent magnet synchronous motor according to an embodiment of the modeling method for permanent magnet synchronous motors based on the intrinsic characteristics of stator and rotor of the present invention. The positive direction of the motor phase current is defined as the direction flowing out of the stator windings (a generator convention). Therefore, the equivalent circuit of the three-phase permanent magnet synchronous motor in the natural coordinate system is as follows: Figure 2 As shown, according to Faraday's law, the stator voltage equation and flux linkage equation of the motor are respectively shown in equation (1) and equation (2):

[0070]

[0071] Λ=L s I-Λ m (2)

[0072] Where E represents the stator induced electromotive force, u = [u a u b u c ]T , representing the stator phase voltage, I = [i a i b i c ] T Λ represents the stator phase current, R represents the stator phase winding resistance, and Λ = [λ]. a λ b λ c ] T , representing stator flux linkage, L represents the permanent magnet flux linkage linked to the stator windings. s This represents the stator inductance matrix.

[0073] For a permanent magnet synchronous motor with three-phase symmetrical, sinusoidal windings, the winding function and the reverse air gap function are obtained by solving with phase a winding as the reference point.

[0074] Different winding arrangements in a motor will generate different magnetomotive forces, thus affecting the distribution of the magnetic field inside the motor. The physical distribution of the windings is generally described using winding functions. According to winding function theory, the self-inductance and mutual inductance components of the stator armature reaction inductance can be represented by the winding function N(γ) and the air gap function.

[0075] The ideal sinusoidal distributed three-phase symmetrical winding is the simplest type of winding. If phase a is taken as the reference point, the winding function waveform diagram of this type of winding is as follows: Figure 3 As shown, for Figure 3 Fourier decomposition of the winding function curve shown can yield the fundamental expression of the winding function shown in equation (3):

[0076]

[0077] Where, N t γ represents the number of turns per phase winding; γ represents the distance from any point in space to the axis of the stator phase a winding.

[0078] In practice, the winding function of a motor often contains winding harmonics. Taking a permanent magnet motor with a symmetrical concentrated full-pitch winding structure as an example, by performing Fourier decomposition on the winding function curve of phase a of the motor stator, the harmonic expression of the winding function shown in equation (4) can be obtained:

[0079]

[0080] Where, N a (γ),N b (γ),N c (γ) are the winding functions of the three-phase windings a, b, and c of the motor, respectively; This represents the winding coefficient.

[0081] For the permanent magnet synchronous motor using a salient pole rotor in the embodiment, its stator and rotor structure is as follows: Figure 1 As shown. The presence of permanent magnets results in a non-uniform air gap width in the motor. The inverse air gap function, the reciprocal of the air gap width, describes the variation of the motor's air gap with spatial position. A schematic diagram of the inverse air gap function waveform for a salient-pole permanent magnet synchronous motor can be approximated using... Figure 4 Let's take the position of the stator phase a axis as the starting point of the spatial position, and obtain the analytical expression of the air gap function of the permanent magnet motor shown in equation (5):

[0082]

[0083] Where, α PM and d PM These represent the width and thickness of the permanent magnet, respectively; d air θ represents the air gap height between the stator and rotor core; θ represents the rotor position angle; k = 2, 4, 6, ..., ∞.

[0084] Based on the fundamental expression of the winding function, the linear stator inductance of the motor is solved with the position of the stator a-phase axis as the starting point of the spatial position.

[0085] Based on the above analysis, the self-inductance and mutual inductance components of the stator armature reaction inductance of the motor can be expressed using the winding function N(γ) and the air gap function g. -1 The expression (γ,θ) is shown in equation (6):

[0086]

[0087] Where the subscripts x and y represent any one of the three-phase stator windings a, b, and c of the motor, if x and y both take the same subscript, L xy μ0 represents the self-inductance of the phase winding, otherwise it represents the mutual inductance of the phase winding; μ0 represents the air permeability; r represents the stator inner diameter; l represents the stator axial length.

[0088] The air gap magnetic flux density curve generated by the rotor magnetic field of the permanent magnet synchronous motor is approximated as a rectangular wave, and the no-load air gap magnetic flux density of the permanent magnet synchronous motor is obtained by solving the problem.

[0089] For the salient-pole permanent magnet synchronous motor involved in this embodiment, due to limitations in the manufacturing process of permanent magnets and rotor assembly, the actual rotor magnetic field is not an ideal sinusoidal distribution, and the rotor magnetic field generated by the permanent magnet has a high harmonic content. A schematic diagram of the air gap magnetic flux density waveform is shown below. Figure 5 As shown. For analytical analysis, the air gap magnetic flux density curve generated by the rotor magnetic field is approximated as a rectangle, that is, simplified to an amplitude of B. δ Width is α PM A rectangular wave (with a permanent magnet width, also known as the calculated pole arc angle) for... Figure 5For any point Γ on the air gap magnetic flux density curve shown, with the rotor axis position (d-axis) as the origin, perform Fourier decomposition to obtain the no-load air gap magnetic flux density of the permanent magnet synchronous motor shown in equation (7):

[0090]

[0091] Among them, B δ α PM These represent the amplitude and width of the approximate rectangular wave, respectively. This represents the amplitude of the kth harmonic of the air gap magnetic flux density of the rotor magnetic field.

[0092] Based on the fundamental expression of the winding function and the no-load air gap magnetic flux density of the permanent magnet synchronous motor, as well as the stator voltage equation and flux linkage equation of the three-phase permanent magnet synchronous motor in the natural coordinate system, the fundamental expression of the permanent magnet flux linkage of the motor rotor is obtained by solving, as shown in equation (8):

[0093]

[0094] in, These represent the permanent magnet flux linkages linked to the three-phase stator windings a, b, and c, respectively; λ m1 =rlπB1N t B1 represents the fundamental amplitude of the permanent magnet flux linkage linked by the phase winding; r represents the fundamental amplitude of the air gap magnetic flux density of the rotor magnetic field; l represents the stator inner diameter; N represents the stator axial length. t n represents the number of turns per phase winding; p This represents the number of rotor pole pairs of the motor.

[0095] For a three-phase symmetrical winding with an ideal sinusoidal distribution, the winding function can be represented by equation (3). Due to the orthogonality of trigonometric functions, the integral of the product of each term after the second term of the inverted air gap function shown in equation (5) with the winding function is zero. The inverted air gap function shown in equation (5) is simplified to equation (9):

[0096] g -1 (γ,θ)=σ0+σ2cos2(γ-θ) (9)

[0097] According to the winding function theory, the self-inductance of phase a of the motor is as shown in equation (10):

[0098]

[0099] At this time, the mutual inductance of the motor's ab phase windings is as shown in equation (11):

[0100]

[0101] Similarly, other armature reaction inductance components can be obtained, and the stator inductance can be represented in matrix form, as shown in equation (12):

[0102]

[0103] In equation (12), the calculation methods for the self-inductance coefficient, leakage inductance coefficient, and mutual inductance coefficients L0, L1, L2 are as shown in equation (13):

[0104]

[0105] Among them, L σ L is the stator leakage inductance matrix; θ For the armature reaction inductance matrix; I 3×3 It is a 3x3 unit diagonal matrix; L0 is the self-inductance coefficient; L1 is the leakage inductance coefficient; L2 is the mutual inductance coefficient; L aa ,L bb ,L cc The self-inductance of the three-phase winding; L ab ,L bc ,L ca The mutual inductance of the three-phase windings is given by μ0; the permeability of air is given by r; the stator inner diameter is given by r; and the stator axial length is given by l. mt N is the length of each turn of the stator winding of the motor; t r is the number of turns per phase of the stator winding. c h is the radius of the copper conductor in the stator winding. ci b is the thickness of the stator winding insulation. w This represents the total thickness of all windings stacked in the stator slot.

[0106] The above analysis shows that for salient-pole motors, the armature reaction inductance matrix only contains constants; for salient-pole motors, due to the salient-pole effect causing the stator inductance to change with the rotor position, the armature reaction inductance matrix adds a term that changes with the rotor position.

[0107] From equations (3), (7), and (8), we can obtain the permanent magnet flux linkage of the stator turns with sinusoidal winding distribution shown in equation (14):

[0108]

[0109] The matrix form of the permanent magnet flux linkage in equation (14) is shown in equation (15):

[0110]

[0111] in, These are the permanent magnet flux linkages linked to the three-phase stator windings a, b, and c, respectively. λ is the amplitude of the first harmonic of the air gap magnetic flux density of the rotor magnetic field; m1 =rlπB1N1 is the amplitude of the first harmonic of the permanent magnet flux linkage linked by the phase winding; the amplitude of the first harmonic of the winding function. N t n is the number of turns per phase winding; p B is the number of rotor pole pairs of the motor; δ This represents the amplitude of the air gap magnetic flux density.

[0112] Based on the fundamental expression of the winding function, the reverse air gap function expression, the no-load air gap magnetic flux density, and the fundamental expression of the permanent magnet flux linkage of the motor rotor, a linear analytical model of the permanent magnet synchronous motor is constructed.

[0113] The constant amplitude PARK transformation matrix is ​​defined in equation (16):

[0114]

[0115] The inverse matrix of the above equation is shown in equation (17):

[0116]

[0117] Applying the PARK transformation to equation (1) yields the voltage equation in the dq coordinates, as shown in equation (18):

[0118]

[0119] Among them, u dq Let u be the stator voltage vector in the dq-axis coordinate system. dq =[u d u q ] T ; u is the stator phase voltage vector in the natural coordinate system, u = [u a u b u c ] T ;u a u b u c These are the three-phase stator voltages a, b, and c, respectively; I dq Let I be the stator current vector in the dq-axis coordinate system. dq =[i d i q ] T I is the stator phase current vector in the natural coordinate system, I = [i a i b i c ] T i a i b i c These represent the three-phase stator currents a, b, and c, respectively; Λ is the stator flux linkage vector in the natural coordinate system, Λ=[λ a λ b λ c ] T;λ a , λ b , λ c The stator flux linkages for phases a, b, and c are respectively; Λ dq Let Λ be the stator flux linkage vector in the dq-axis coordinate system. dq =[λ d λ q ] T .

[0120] Calculate each component in equation (18) as shown in equations (19)-(21):

[0121]

[0122]

[0123]

[0124] The dq-axis component of the permanent magnet flux linkage linked with the stator flux linkage is shown in equation (22):

[0125] Λ mdq =C 32 (θ)Λ m =[λ m1 0] T (twenty two)

[0126] Combining equations (17) and (22), we obtain the linear analytical model of the permanent magnet synchronous motor shown in equation (23):

[0127]

[0128] The stator flux linkage equation and the dq-axis inductance equation are shown in equations (24) and (25), respectively:

[0129]

[0130]

[0131] Where L0, L1, and L2 are the self-inductance, leakage inductance, and mutual inductance, respectively, u d ,u q i d i q These represent the stator voltage and current in the dq coordinate system, respectively, and λ d ,λ q Let L be the stator flux linkage in the dq coordinate system. d ,L q λ is the stator inductance along the dq axis; m1 ω is the amplitude of the first harmonic of the permanent magnet flux linkage in each phase winding of the stator. e This refers to the electrical angular velocity of the motor rotor.

[0132] The nonlinear modeling method for permanent magnet synchronous motors based on the intrinsic characteristics of stator and rotor according to the second embodiment of the present invention is described in detail below, based on the first embodiment:

[0133] Due to limitations in manufacturing processes, the winding function of an actual motor will contain winding harmonics. This step will establish a nonlinear mathematical model of a permanent magnet synchronous motor containing winding harmonics. The amplitude of the kth harmonic of the winding function represented by Equation (4) decays rapidly at a rate that is a multiple of k. For the sake of simplicity, only the fundamental wave and the 3rd and 5th harmonics in the winding function are considered, and the influence of other higher harmonics is ignored. Equation (4) can be transformed into Equation (26):

[0134]

[0135] in, N t n is the number of turns per phase winding; p This represents the number of rotor pole pairs of the motor.

[0136] Due to the orthogonality of trigonometric functions, the integral of the product of the 12th and higher harmonics of the inverted air gap function shown in equation (5) and the winding function shown in equation (26) is zero. Using the above property, equation (26) can be transformed into equation (27):

[0137]

[0138] in, k = 2, 4, 6, 8, 10; α PM and d PM These represent the width and thickness of the permanent magnet, respectively; d air θ is the air gap height between the stator and rotor cores; θ is the rotor position angle.

[0139] The three-phase self-inductances of the permanent magnet synchronous motor obtained from equation (6) are shown in equations (28) to (30):

[0140]

[0141]

[0142]

[0143] Similarly, the three mutual inductances are obtained as shown in equations (31)-(33):

[0144]

[0145]

[0146]

[0147] In the phase coordinate system, the stator inductance matrix can be expressed by the intrinsic parameters of the intrinsic motor as equation (34):

[0148]

[0149] Among them, L σ L is the stator leakage inductance matrix; θ For the armature reaction inductance matrix; I 3×3 It is a 3x3 unit diagonal matrix; L0 is the self-inductance coefficient; L1 is the leakage inductance coefficient; L2 is the mutual inductance coefficient; L aa ,L bb ,L cc The self-inductance of the three-phase winding; L ab ,L bc ,L ca It is the mutual inductance of the three-phase windings.

[0150] According to the winding function theory, the permanent magnet flux linkage of the three-phase windings of the permanent magnet synchronous motor can be expressed as equation (35).

[0151]

[0152] Rewrite the above equation as a matrix of equation (36):

[0153]

[0154] in, These are the permanent magnet flux linkages linked to the three-phase windings a, b, and c, respectively; the amplitude of the k-th harmonic of the permanent magnet flux linkage linked to the motor phase windings is λ. mk , λ mk =rlπB k N k .

[0155] From equations (1), (16), (17), (35), and (36), we can obtain the dq-axis voltage equation considering the influence of two main nonlinear factors: the non-sinusoidal distribution of the stator winding and the rotor magnetic field harmonics of the permanent magnet synchronous motor. This is the nonlinear analytical model of the permanent magnet synchronous motor, as shown in equation (37).

[0156]

[0157] Among them, u dq Let u be the stator voltage vector in the dq-axis coordinate system. dq =[u d u q ] T ;I dq Let I be the stator current vector in the dq-axis coordinate system. dq =[i d iq ] T Λ dq Let Λ be the stator flux linkage vector in the dq-axis coordinate system. dq =[λ d λ q ] T ;ω e λ is the electrical angular velocity of the motor rotor. m1 ,λ m5 These are the amplitudes of the first and fifth harmonics of the permanent magnet flux linkage linked to the motor windings, respectively.

[0158] The formula for calculating the coefficient related to inductance is shown in equation (38):

[0159]

[0160] Based on the above analysis, the linear and nonlinear models of the permanent magnet synchronous motor in analytical form, expressed by the intrinsic parameters of the motor, are finally obtained: the linear analytical model of the permanent magnet synchronous motor based on the three-phase symmetrical sinusoidal distribution winding (23), and the nonlinear analytical model of the permanent magnet synchronous motor considering the influence of nonlinear factors such as the non-sinusoidal distribution of the stator winding and the harmonics of the rotor magnetic field (37); the relevant parameters in the model can be accurately obtained by calculation from the intrinsic parameters of the motor.

[0161] Although the steps in the above embodiments are described in the above order, those skilled in the art will understand that in order to achieve the effect of this embodiment, different steps do not need to be executed in such an order. They can be executed simultaneously (in parallel) or in a reverse order. These simple variations are all within the protection scope of this invention.

[0162] The permanent magnet synchronous motor modeling system based on the intrinsic characteristics of stator and rotor according to the third embodiment of the present invention includes:

[0163] The equation construction module is configured to construct the stator voltage equation and flux linkage equation of a three-phase permanent magnet synchronous motor in the natural coordinate system based on Faraday's law.

[0164] The winding function and reverse air gap function solving module is configured to, for a permanent magnet synchronous motor with three-phase symmetrical sinusoidal windings, use the a-phase winding as the reference point to solve for the fundamental wave expression and reverse air gap function expression of the winding function; for a permanent magnet synchronous motor with non-sinusoidal winding distribution, use the a-phase winding as the reference point to solve for the harmonic expression of the winding function.

[0165] The stator inductance solving module is configured to solve the linear stator inductance of the motor based on the fundamental expression of the winding function, taking the position of the stator a-phase axis as the starting point of the spatial position; and to solve the nonlinear stator inductance of the motor based on the harmonic expression of the winding function.

[0166] The no-load air gap magnetic flux density calculation module is configured to approximate the air gap magnetic flux density curve generated by the rotor magnetic field of the permanent magnet synchronous motor as a rectangular wave, and solve for the no-load air gap magnetic flux density of the permanent magnet synchronous motor.

[0167] The permanent magnet flux linkage solving module is configured to solve for the fundamental expression of the permanent magnet flux linkage of the motor rotor based on the fundamental expression of the winding function, the no-load air gap magnetic flux density of the permanent magnet synchronous motor, and the stator voltage equation and flux linkage equation of the three-phase permanent magnet synchronous motor in the natural coordinate system; and to solve for the harmonic expression of the permanent magnet flux linkage of the motor rotor based on the harmonic expression of the winding function, the no-load air gap magnetic flux density of the permanent magnet synchronous motor, and the stator voltage equation and flux linkage equation of the three-phase permanent magnet synchronous motor in the natural coordinate system.

[0168] The model building module is configured to construct a linear analytical model of the permanent magnet synchronous motor based on the fundamental expression of the winding function, the inverted air gap function expression, the unloaded air gap magnetic flux density, and the fundamental expression of the permanent magnet flux linkage of the motor rotor; and to construct a nonlinear analytical model of the permanent magnet synchronous motor based on the harmonic expression of the winding function, the inverted air gap function expression, the unloaded air gap magnetic flux density, and the harmonic expression of the permanent magnet flux linkage of the motor rotor.

[0169] Those skilled in the art will understand that, for the sake of convenience and brevity, the specific working process and related descriptions of the system described above can be found in the corresponding processes in the foregoing method embodiments, and will not be repeated here.

[0170] It should be noted that the permanent magnet synchronous motor modeling system based on the intrinsic characteristics of the stator and rotor provided in the above embodiments is only an example of the division of the above functional modules. In practical applications, the above functions can be assigned to different functional modules as needed, that is, the modules or steps in the embodiments of the present invention can be further decomposed or combined. For example, the modules in the above embodiments can be merged into one module, or further divided into multiple sub-modules to complete all or part of the functions described above. The names of the modules and steps involved in the embodiments of the present invention are only for distinguishing the various modules or steps and are not considered as an improper limitation of the present invention.

[0171] An electronic device according to a fourth embodiment of the present invention includes:

[0172] At least one processor; and

[0173] A memory communicatively connected to at least one of the processors; wherein,

[0174] The memory stores instructions that can be executed by the processor to implement the above-described modeling method for permanent magnet synchronous motors based on the intrinsic characteristics of stator and rotor.

[0175] A computer-readable storage medium according to a fifth embodiment of the present invention stores computer instructions, which are executed by the computer to implement the above-described modeling method for permanent magnet synchronous motors based on the intrinsic characteristics of stator and rotor.

[0176] Those skilled in the art will understand that, for the sake of convenience and brevity, the specific working process and related descriptions of the storage device and processing device described above can be referred to the corresponding process in the foregoing method embodiments, and will not be repeated here.

[0177] Those skilled in the art will recognize that the modules and method steps of the various examples described in conjunction with the embodiments disclosed herein can be implemented in electronic hardware, computer software, or a combination of both. The programs corresponding to the software modules and method steps can be placed in random access memory (RAM), main memory, read-only memory (ROM), electrically programmable ROM, electrically erasable programmable ROM, registers, hard disks, removable disks, CD-ROMs, or any other form of storage medium known in the art. To clearly illustrate the interchangeability of electronic hardware and software, the components and steps of the various examples have been generally described in terms of functionality in the foregoing description. Whether these functions are implemented in electronic hardware or software depends on the specific application and design constraints of the technical solution. Those skilled in the art can use different methods to implement the described functions for each specific application, but such implementation should not be considered beyond the scope of the invention.

[0178] The terms “first”, “second”, etc., are used to distinguish similar objects, not to describe or indicate a specific order or sequence.

[0179] The term "comprising" or any other similar term is intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus / device that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent in such process, method, article, or apparatus / device.

[0180] The technical solution of the present invention has been described above with reference to the preferred embodiments shown in the accompanying drawings. However, it will be readily understood by those skilled in the art that the scope of protection of the present invention is obviously not limited to these specific embodiments. Without departing from the principles of the present invention, those skilled in the art can make equivalent changes or substitutions to the relevant technical features, and the technical solutions after these changes or substitutions will all fall within the scope of protection of the present invention.

Claims

1. A method for modeling a permanent magnet synchronous motor based on the eigen characteristics of the stator and rotor, characterized in that, The permanent magnet synchronous motor modeling method comprises the following steps: In step S10, a stator voltage equation and a flux linkage equation of a three-phase permanent magnet synchronous motor in a natural coordinate system are constructed based on Faraday's law; In step S20, for a permanent magnet synchronous motor with three-phase symmetrical and sinusoidal distribution windings, a fundamental wave expression and a back air gap function expression of a winding function are obtained by taking the a-phase winding as a reference point; for a permanent magnet synchronous motor with non-sinusoidal distribution windings, a harmonic wave expression of the winding function is obtained by taking the a-phase winding as the reference point; In step S30, based on the fundamental wave expression of the winding function, a linear stator inductance of the motor is solved by taking the stator a-phase axis position as the starting point of the spatial position; based on the harmonic wave expression of the winding function, a nonlinear stator inductance of the motor is solved; In step S40, an air gap flux density curve generated by a rotor magnetic field of the permanent magnet synchronous motor is approximated as a rectangular wave, and a no-load air gap flux density of the permanent magnet synchronous motor is solved; In step S50, based on the fundamental wave expression of the winding function and the no-load air gap flux density of the permanent magnet synchronous motor, and the stator voltage equation and the flux linkage equation of the three-phase permanent magnet synchronous motor in the natural coordinate system, a fundamental wave expression of a rotor permanent magnet flux linkage of the motor is solved; based on the harmonic wave expression of the winding function and the no-load air gap flux density of the permanent magnet synchronous motor, and the stator voltage equation and the flux linkage equation of the three-phase permanent magnet synchronous motor in the natural coordinate system, a harmonic wave expression of the rotor permanent magnet flux linkage of the motor is solved; In step S60, based on the fundamental wave expression of the winding function, the back air gap function expression, the no-load air gap flux density, and the fundamental wave expression of the rotor permanent magnet flux linkage of the motor, a linear analytical model of the permanent magnet synchronous motor is constructed; based on the harmonic wave expression of the winding function, the back air gap function expression, the no-load air gap flux density, and the harmonic wave expression of the rotor permanent magnet flux linkage of the motor, a nonlinear analytical model of the permanent magnet synchronous motor is constructed.

2. The method of claim 1, wherein, The stator voltage equation and the flux linkage equation of the three-phase permanent magnet synchronous motor in the natural coordinate system are expressed as: Λ = L s I - Λ m where E represents the stator induced electromotive force; u = [u a u b u c ] T represents the stator phase voltage; I = [i a i b i c ] T represents the stator phase current; R represents the resistance of the stator phase winding; Λ = [λ a λ b λ c ] T represents the stator flux linkage; represents the permanent magnet flux linkage linked with the stator winding; L s represents the stator inductance matrix.

3. The method of claim 1, wherein, The fundamental wave expression and the back air gap function expression of the winding function are expressed as: where N t represents the number of turns of each phase winding; γ represents the distance from an arbitrary point in space to the axis of the stator a-phase winding; α PM and d PM represent the width and thickness of the permanent magnet, respectively; d air represents the air gap height of the stator and rotor cores; θ represents the rotor position angle; k = 2, 4, 6, …, ∞.

4. The method of claim 3, wherein, The harmonic wave expression of the winding function is expressed as: wherein represents the winding coefficient.

5. The method of claim 4, wherein, The no-load air gap flux density of the permanent magnet synchronous motor is expressed as: wherein B δ , a PM are the amplitude and width of the approximated rectangular wave, respectively, is the amplitude of the kth harmonic of the rotor magnetic field air-gap flux density.

6. The method of claim 5, wherein, The fundamental wave expression of the rotor permanent magnet flux linkage of the motor is expressed as: wherein λa, λb, λc represent the permanent magnet flux linkage per turn of the three-phase stator windings a, b, c, respectively; λ m1 = rπB1N t B1 represents the fundamental amplitude of the rotor magnetic field air-gap flux density; r represents the inner diameter of the stator; l represents the axial length of the stator; N t represents the number of turns of each phase winding; n p represents the number of pairs of rotor magnetic poles of the motor.

7. The method of claim 6, wherein, The harmonic wave expression of the rotor permanent magnet flux linkage of the motor is expressed as: where λ mk = rπB k N t is the amplitude of the kth harmonic of the permanent magnet flux linkage per turn of the phase winding; B k represents the amplitude of the kth harmonic of the rotor field air gap flux density; r represents the inner diameter of the stator; l represents the axial length of the stator; N t represents the number of turns of the phase winding; n p represents the number of pairs of rotor magnetic poles of the motor.

8. The method of claim 7, wherein, The linear analytical model of the permanent magnet synchronous motor is expressed as: Wherein, L0, L1, L2 represent self-inductance, leakage inductance and mutual inductance respectively; u d , q , d , q represent stator voltage and current in dq coordinate system respectively; λ d , q represent stator flux linkage in dq coordinate system; L d , q represent dq-axis stator inductance; λ m1 represent the fundamental amplitude of the permanent magnet flux linkage of each phase winding of the stator; ω e represent the electrical angular velocity of the motor rotor.

9. The method of claim 8, wherein, The self-inductance coefficient, the leakage inductance coefficient, and the mutual inductance coefficient L0, L1, and L2 are expressed as: wherein μ0 represents the air permeability, l mt represents the length of each turn of the motor stator winding; r c represents the radius of the stator winding wire; h ci represents the insulation thickness of the stator winding; b w represents the total thickness of all winding layers in the stator slot.

10. The method of claim 8, wherein, The nonlinear analytical model of the permanent magnet synchronous motor is expressed as: where, u dq = [u d u q ] T , I dq = [i d i q ] T , u d , u q , i d , i q represent stator voltage, current in dq coordinate system respectively; λ m5 represents the fifth harmonic amplitude of the permanent magnet flux linkage of each phase winding of the stator per turn; σ2, σ4, σ6, σ 10 represent the 2, 4, 6, 10 order coefficients of the inverse air gap function respectively; N1, N5 represent the first and fifth order coefficients of the winding function respectively.

Images