Permanent magnet synchronous motor zero-order radial force analysis modeling method based on lumped parameters

Abstract

The invention discloses a permanent magnet synchronous motor zero-order radial force analysis modeling method based on lumped parameters, and belongs to the technical field of motor electromagnetic vibration noise analysis. According to the method, the flux linkage is taken as a key correlation variable, and the relationship between the air gap flux density distribution and the current is revealed by reasonably selecting lumped parameters. The relationship between current and flux linkage is established by using incremental inductance, permanent magnet flux linkage and armature flux linkage are separated by using apparent inductance, and the influence of flux leakage on air gap armature flux linkage is analyzed by using stator leakage inductance. According to a winding structure, an air gap flux density distribution basic waveform is established, and the armature air gap flux density distribution characteristic is further refined by analyzing the influence of a stator structure and a magnetic bridge. In addition to traditional lumped parameters, new lumped parameters are also defined for establishing an analytical model. Compared with a finite element model in the prior art, the analytical model disclosed by the invention has higher precision, and can intuitively express the relationship among the current, the lumped parameter and the zero-order radial electromagnetic force.

Description

Technical Field

[0001] The present invention relates to the technical field of electromagnetic vibration and noise analysis of permanent magnet synchronous motors, and particularly to an analytical modeling method for zero-order radial force of permanent magnet synchronous motors based on lumped parameters. Background Art

[0002] The vigorous promotion of new energy electric vehicles helps to get rid of the dependence on fossil energy and is of great significance for alleviating the energy and environmental crises. As a core component of electric vehicles, the permanent magnet synchronous motor (PMSM) has attracted much attention due to its various types and excellent performance. With the rapid development of the new energy vehicle industry in recent years, the performance requirements for permanent magnet synchronous motors are getting higher and higher, such as higher power density, smoother torque, and lower noise, vibration, and harshness (NVH). Many researchers at home and abroad are committed to improving the performance of permanent magnet synchronous motors by designing control methods.

[0003] Electromagnetic vibration and noise are one of the main noise sources of electric vehicles. Integer-slot permanent magnet synchronous motors usually have few slots and are mostly used in centralized drive systems of electric vehicles. The lowest non-zero spatial order of the radial electromagnetic force wave of this type of motor is relatively high. According to the relationship between vibration intensity and spatial order, the influence of these high spatial order forces on vibration is very small and is usually ignored. For integer-slot interior permanent magnet synchronous motors, the zero-order radial electromagnetic force is the main cause of vibration and noise. Electromagnetic vibration is mainly caused by stator resonance excited by radial electromagnetic force waves, especially when the spatial order and frequency of the radial electromagnetic force are consistent with the resonance frequency of the corresponding stator mode.

[0004] The radial electromagnetic force models of permanent magnet synchronous motors are mainly divided into numerical models and analytical models. The numerical model refers to calculating the electromagnetic force using the Maxwell stress tensor equation based on the air-gap flux density obtained from finite element analysis. Although this model has high accuracy, it cannot reveal the relationship between current and radial electromagnetic force. Analytical models are mostly based on magnetic circuit principles and establish analytical models using magnetomotive force and permeability considering tooth-slot structures. Although this model can reveal the relationship between current and radial electromagnetic force, its calculation accuracy is relatively low and it is usually applicable to qualitative analysis.

[0005] In the research of many current scholars, the relationship between radial electromagnetic force and current harmonics is mostly qualitative analysis, and no expression for quantitative description has been formed. The analytical model lacks sufficient consideration of magnetic saturation. A radial electromagnetic force model explicitly containing current harmonics has not been established. Summary of the Invention

[0006] The present invention aims at the problem of electromagnetic vibration analysis of an interior permanent magnet synchronous motor with integral-slot windings, and provides an analytical modeling method for the zero-order radial force of a permanent magnet synchronous motor based on lumped parameters. This analytical method can intuitively express the relationship between current, lumped parameters and zero-order radial electromagnetic force, and has higher accuracy compared with the finite element model.

[0007] To solve the above technical problems, the present invention provides the following technical solutions:

[0008] An analytical modeling method for the zero-order radial force of a permanent magnet synchronous motor based on lumped parameters, which is used for an interior permanent magnet synchronous motor with integral-slot windings, includes:

[0009] Step (a): Select the magnetic flux linkage as the key correlation variable to analyze the relationship between the air-gap magnetic flux density distribution and the current. Among them, the types of magnetic flux linkage are divided into winding magnetic flux linkage and air-gap magnetic flux linkage according to the observation position, and into armature magnetic flux linkage and permanent magnet magnetic flux linkage according to the generation cause;

[0010] Step (b): Use the incremental inductance to establish the relationship between current and magnetic flux linkage when considering current harmonics, use the apparent inductance to separate the permanent magnet magnetic flux linkage and the armature magnetic flux linkage, and use the stator leakage inductance to analyze the influence of leakage flux on the air-gap magnetic flux linkage;

[0011] Step (c): Establish the basic waveform of the air-gap magnetic flux density distribution according to the winding structure, and further refine the air-gap magnetic flux density distribution characteristics considering the influence of the stator structure and the magnetic bridge;

[0012] Step (d): By analyzing the relationship between the air-gap magnetic flux density and the air-gap magnetic flux linkage, establish the relationship between the lumped parameters, current and the air-gap armature magnetic flux density distribution;

[0013] Step (e): Use the method of freezing permeability to extract the air-gap permanent magnet magnetic flux density, define the lumped parameters C, D, E related to the stator-rotor structure and the air-gap permanent magnet magnetic flux density distribution, and establish an analytical model of the zero-order radial electromagnetic force explicitly containing current.

[0014] The present invention has the following beneficial effects:

[0015] (1) The model explicitly contains current and can intuitively express the relationship between current, lumped parameters and zero-order radial force;

[0016] (2) It fully considers the influence of magnetic field saturation and stator-rotor structure and has higher accuracy;

[0017] (3) The zero-order radial electromagnetic force still has high accuracy when considering current harmonics, laying a foundation for injecting harmonics to suppress electromagnetic vibration noise. Description of the Drawings

[0018] Figure 1 It is the electromagnetic structure diagram of a certain integral-slot interior permanent magnet synchronous motor;

[0019] Figure 2(a) is a definition diagram of the apparent inductance and incremental inductance when the permanent magnet and the armature are simultaneously excited;

[0020] Figure 2(b) is a comparison diagram of the flux linkage errors in the calculation of the apparent inductance and incremental inductance when considering current harmonics;

[0021] Figure 3(a) is for the rotor mechanical angle θ r and the spatial angle definition diagram;

[0022] Figure 3(b) is a schematic diagram of the air-gap armature magnetic flux density distribution when considering the winding structure and ignoring the influence of the stator and rotor structures;

[0023] Figure 4(a) is a schematic diagram for the analysis of the magnetic flux density distribution at the magnetic bridge;

[0024] Figure 4(b) is for the rotor mechanical angle θ r = 0°, the spatial distribution function diagrams of m s and m r under a certain current excitation;

[0025] Figure 5(a) is for θ e = 10°, the air-gap magnetic flux density distribution diagrams of the analytical model and the finite element model when the fundamental amplitude and phase of the current are (30A, 107.7°);

[0026] Figure 5(b) is for θ e = 10°, the air-gap magnetic flux density distribution diagrams of the analytical model and the finite element model when the fundamental amplitude and phase of the current are (60A, 115.2°);

[0027] Figure 6 is the flow chart of the lumped parameter extraction strategy based on finite element analysis;

[0028] Figures 7(a), (b), (c), (d) are respectively the lumped parameters C, D, E, F pm 0 when the fundamental amplitude and phase of the current are (30A, 107.7°);

[0029] Figures 8(a), (b), (c), (d) are respectively the lumped parameters C, D, E, F pm 0 when the fundamental amplitude and phase of the current are (60A, 115.2°);

[0030] Figure 9 is the comparison diagram of the analytical model and the finite element calculation results of F r0 under sinusoidal excitation and harmonic excitation;

[0031] Figure 10 is the schematic flow diagram of the zero-order radial force analytical modeling method for the permanent magnet synchronous motor based on lumped parameters of the present invention. Detailed implementation manners

[0032] To make the technical problems, technical solutions and advantages to be solved by the present invention clearer, the following will be described in detail with reference to the accompanying drawings and specific embodiments.

[0033] The present invention mainly aims at an interior permanent magnet synchronous motor with integral-slot distributed windings. For the convenience of description, a six-phase permanent magnet synchronous motor for vehicles is taken as an example for illustration. The basic parameters of the motor are shown in Table 1, and the electromagnetic structure is as Figure 1 shown. The motor winding adopts a short-pitch winding, the number of winding turns is 12, and the number of parallel branches is 4; reference numeral 11 is the iron core, reference numeral 12 is the winding, and reference numeral 13 is the permanent magnet.

[0034] Table 1 Parameters of the permanent magnet synchronous motor

[0035]

[0036] The zero-order radial electromagnetic force is the main cause of the stator vibration of the integral-slot winding interior permanent magnet synchronous motor. The present invention aims to establish an analytical model of the zero-order radial electromagnetic force that explicitly includes current.

[0037] The present invention provides an analytical modeling method for the zero-order radial force of a permanent magnet synchronous motor based on lumped parameters, which is used for an integral-slot winding interior permanent magnet synchronous motor, as Figure 10 shown, and includes:

[0038] Step (a): Select the magnetic flux as the key correlation variable to analyze the relationship between the air-gap magnetic density distribution and the current. Among them, the magnetic flux types are divided into winding magnetic flux and air-gap magnetic flux according to the observation position, and into armature magnetic flux and permanent-magnet magnetic flux according to the generation cause;

[0039] In this step, the magnetic flux is selected as the key correlation variable to analyze the relationship between the air-gap magnetic density distribution and the current in the subsequent analysis, and at the same time to establish the relationship between the current, lumped parameters, magnetic density distribution and radial electromagnetic force. The magnetic flux not only reflects the relationship between voltage, current and torque, but also is related to the magnetic field distribution, and is the key to connecting the magnetic field and the current.

[0040] Step (b): Use the incremental inductance to establish the relationship between the current and the magnetic flux when considering current harmonics, use the apparent inductance to separate the permanent-magnet magnetic flux and the armature magnetic flux, and use the stator leakage inductance to analyze the influence of leakage flux on the air-gap magnetic flux;

[0041] In this step, the apparent inductance and the incremental inductance are synchronously selected and reasonably used. The apparent inductance can effectively separate the permanent-magnet magnetic flux and the armature magnetic flux, and the incremental inductance effectively broadens the local linearization region of the magnetic flux.

[0042] As shown in Fig. 2(a), according to the definition of the apparent inductance The total magnetic flux linkage ψ can be established L and the current i L , armature magnetic flux linkage , permanent magnet magnetic flux linkage The relational expression between Similarly, according to the definition of incremental inductance L inc = dψ L / di L , the total magnetic flux linkage ψ can be established L and the current i L , the virtual permanent magnet magnetic flux linkage in the case of incremental inductance The relationship between Here, the virtual permanent magnet magnetic flux linkage only reflects the numerical relationship and does not represent the permanent magnet magnetic flux linkage in the actual physical sense.

[0043] According to the definition of incremental inductance, the incremental inductance can expand the local linearization range and improve the calculation accuracy of the magnetic flux linkage when considering current harmonics. As shown in Fig. 2(b), when the current changes by Δi L , the change in the magnetic flux linkage calculated by the apparent inductance compared with the change in the magnetic flux linkage calculated by the incremental inductance has a large error. Therefore, the incremental inductance has great advantages in calculating the magnetic flux linkage (especially when considering current harmonics). The apparent inductance can be used to separate the winding magnetic flux generated by the permanent magnet and the armature.

[0044] In this step, when considering each phase winding of the motor comprehensively, the magnetic flux linkage relational expressions all become vector relational expressions, as shown in the following formula:

[0045]

[0046] Among them, the total magnetic flux linkage ψ = [ψ a1 ψ a2 ψ b1 ψ b2 ψ c1 ψ c2 T , a1, a2, b1, b2, c1, c2 are the numbers of each phase winding of the motor, and ψ a1 to ψ c2 are the magnetic flux linkages of each phase winding; is the virtual armature magnetic flux linkage in the case of incremental inductance; is the virtual permanent magnet magnetic flux linkage in the case of incremental inductance; L inc is the incremental inductance matrix, which can be directly obtained through finite element analysis; the phase current i = [i a1 i a2 i b1 i b2 i c1 i c2 T ​​; the a1-phase current i a1 = i 1 cos(θ e + α 1 ), θ e = n p θ r is the rotor electrical angle, n p is the number of pole pairs of the motor, n r is the rotor mechanical angle; i 1 and α 1 are the fundamental wave current amplitude and phase; the definitions of the remaining phase currents are the same and will not be elaborated here;

[0047] Incremental inductance matrix

[0048] where, l a1a1 represents the self-inductance of the a1-phase under the condition of incremental inductance, l a1a2 represents the mutual inductance between the a1-phase and the a2-phase under the condition of incremental inductance; the definitions of the remaining parameters are the same and will not be elaborated here;

[0049] The armature flux linkage and the permanent magnet flux linkage can be separated by using the apparent inductance, so we can get:

[0050]

[0051] where, is the armature flux linkage under the condition of apparent inductance, is the permanent magnet flux linkage under the condition of apparent inductance;

[0052] In practice, not all of the flux linkage generated by the motor winding passes through the air gap and enters the rotor, but a part of it passes through the teeth and slots to form a closed loop, as shown by the dotted line in Figure 4(a). In order to fully consider this part of the stator leakage flux and improve the calculation accuracy of the air gap magnetic density, in the present invention, the stator leakage inductance is used to analyze the leakage flux effect. After considering the stator leakage flux, the part of the armature flux linkage passing through the air gap, that is, the air gap armature flux linkage where, represents the a1-phase air gap flux linkage, and the definitions of the remaining parameters are the same and will not be elaborated here.

[0053] After considering the stator leakage flux, according to (I) and (II), the air gap armature flux linkage is shown in the following formula:

[0054]

[0055] Step (c): Establish the basic waveform of the air gap magnetic density distribution according to the winding structure, and further refine the characteristics of the air gap magnetic density distribution considering the influence of the stator structure and the magnetic bridge;

[0056] In this step, coefficient modeling considering the influence of the stator and rotor structures on the magnetic flux density distribution is carried out. The lumped parameters are winding parameters that vary with the rotor position (or time). The magnetic flux density (referred to as magnetic density) distribution is the spatial distribution characteristic of the rotor at a certain position (or a certain moment). Analyzing the relationship between the air-gap armature magnetic density, permanent magnet magnetic density and lumped parameters at a certain rotor position is the key to obtaining an analytical model, where the magnetic flux linkage is the key variable for establishing the connection. The total air-gap magnetic flux density B g is composed of the air-gap armature magnetic density B am and the air-gap permanent magnet magnetic density B pm .

[0057] As shown in the positional relationship in Fig. 3(a), at the rotor mechanical angle of θ am and B pm , the spatial distribution relationships varying with the spatial angle r are respectively , where A is an arbitrary point in the air gap. When the influence of the stator slot structure and rotor structure is not considered, the magnetic flux density generated by each winding is distributed in a square wave form along the air gap. If all windings are considered, the air-gap armature magnetic density presents a stepped wave distribution along the air gap, denoted as , as shown by the blue line in Fig. 3(b). As shown in Fig. 3(b).

[0058] In the case of high magnetic saturation, the stator and rotor structures have a greater influence on the armature magnetic density distribution. In order to obtain a more accurate armature magnetic flux density distribution, the stator distribution coefficient m d and the rotor distribution coefficient m r are defined in the present invention. The calculation of the stator distribution coefficient m s can refer to the calculation method in Ref. [1] (Z. Zhu, D. Howe, “Analytical prediction of the cogging torque in radial-field permanent magnet brushless motors,” IEEE Transactions on Magnetics, vol. 28, no. 2, pp. 1371-1374, Mar. 1992, DOI: 10.1109 / 20.123947.). When analyzing the rotor distribution coefficient m r , considering that the magnetic saturation at the magnetic bridge of the interior permanent magnet synchronous motor is relatively large and the air-gap magnetic permeability is relatively low, the magnetic resistance of these two parts is relatively large, which has a greater influence on the magnetic flux density distribution near the magnetic bridge. In order to analyze the air-gap magnetic density distribution at the magnetic bridge, two magnetic path channels are selected for analysis, as shown in Fig. 4(a). The magnetic resistances at the magnetic bridge and air gap in the channel are R b and R g respectively, which can be calculated by the following formula, and the structural parameters of each part are shown in Fig. 4(a).

[0059]

[0060] Among them, μ 0 is the vacuum permeability; μ rb is the relative permeability of the magnetic bridge; L g is the air-gap length; l b is the magnetic bridge length; l m is the axial length of the motor; w p is the width of the magnetic circuit channel.

[0061] The reluctances in different magnetic circuit channels are different, so the m r values at different spatial angles are different. Assuming that at the magnetic bridge, m r varies continuously and linearly along l b , m rh and m rl respectively represent the larger and smaller values of m r , and their relationship satisfies the following formula:

[0062]

[0063] Among them, is the spatial electrical angle; m r also needs to satisfy

[0064] According to the method in the present invention, for this vehicle permanent magnet synchronous motor, when θ r = 0°, under a certain current excitation, the distribution characteristics of m s and m r are shown in Fig. 4(b).

[0065] Step (d): By analyzing the relationship between the air-gap magnetic density and the air-gap magnetic flux linkage, establish the relationship between the lumped parameters, the current and the distribution of the air-gap armature magnetic density;

[0066] In this step, the preliminary relationship between the lumped parameters and the distribution parameters is constructed. According to the winding structure and the initial stepped distribution characteristics of the air-gap armature magnetic density, the expression of the relationship between the distribution of the air-gap armature magnetic density and the current can be obtained.

[0067] Specifically, by analyzing the relationship between the air-gap magnetic density and the air-gap magnetic flux linkage, the relationship between the current and the air-gap armature magnetic density is established. Considering the stator-rotor influence coefficients m s and m r and taking the air-gap magnetic flux linkage of phase a 1 as an example, according to the relationship shown in Fig. 4(a), the magnetic flux is related to the integral of the magnetic density distribution, and the following formula can be deduced.

[0068]

[0069] where N is the number of turns of the winding; r is the air-gap radius; to is the spatial angle corresponding to the nth tooth on the stator.

[0070] According to the symmetry of the magnetic flux density distribution, (VI) can be simplified to the following formula:

[0071]

[0072] Considering the stepped distribution characteristic of, (VII) can be simplified to the following formula after being expanded by angle:

[0073]

[0074] where n = 1...5; B t1 …B t5 As shown in Fig. 3(b), the air-gap armature magnetic flux density is the amplitude of each segment of the stepped wave.

[0075] Define the air-gap armature magnetic flux density vector Taking into account the six-phase winding situation comprehensively, the following formula can be obtained:

[0076]

[0077] where A c is the winding coefficient matrix, C = diag([c 1 c 2 c 3 c 4 c 5 c 6 ).

[0078] (IX) establishes the relationship between the lumped parameters, current and air-gap armature magnetic flux density distribution. Considering the stator-rotor influence coefficients m s and m r after that, the actual air-gap magnetic flux density To prove the effectiveness of the analysis method, when the fundamental current amplitude and phase are (30 A, 107.7°) and (60 A, 115.2°) respectively, the air-gap magnetic flux density distribution calculated according to the lumped parameters is shown in Fig. 5.

[0079] Step (e): Extract the air-gap permanent-magnet magnetic flux density by using the method of freezing the magnetic permeability, define the lumped parameters C, D, E related to the stator-rotor structure and the air-gap permanent-magnet magnetic flux density distribution, and establish an analytical model of the zero-order radial electromagnetic force explicitly containing current.

[0080] In addition to the armature magnetic flux density, the permanent magnet magnetic flux density is also an important factor affecting the radial electromagnetic force. In this step, the permanent magnet magnetic flux density can be directly obtained by using the method of freezing the magnetic permeability in finite element analysis, and through appropriate derivation, it is transformed into the lumped parameters C, D, and E required for the lumped parameter analytical model, and then an analytical model of the zero-order radial electromagnetic force explicitly containing current is established.

[0081] Air-gap magnetic flux density B g = B am + B pm , according to the Maxwell tensor equation, the radial electromagnetic force density f r is shown as follows:

[0082]

[0083] where μ 0 is the vacuum magnetic permeability. Integrating f r along the spatial angle gives the zero-order radial electromagnetic force density F r0 , as shown in the following equation:

[0084]

[0085] (XI) The zero-order radial electromagnetic force density F r0 consists of three parts. Among them, the part defined as being generated solely by B pm is F pm0 , and this part can be directly calculated from B pm obtained by freezing the magnetic permeability, as shown in the following equation:

[0086]

[0087] Therefore, the expression (XI) of F r0 can be transformed into the following equation:

[0088]

[0089] where,

[0090] Further simplifying (XIII), we get:

[0091]

[0092] where, D = diag([d 1 d 2 d 3 d 4 d 5 d 6 ); E = diag([e 1 e 2 e 3 e4 e 5 e 6 )。

[0093] Substituting (IX) into (XIV), an analytical expression between the zero-order radial electromagnetic force density and the lumped parameters is obtained as shown below:

[0094]

[0095] where

[0096] (XV) is the established zero-order radial electromagnetic force analytical model, in which the phase current i is explicitly included, intuitively showing the relationship between F r0 and i. Therefore, this model is of great significance for analyzing the influence of current harmonics on F r0 . The lumped parameters of this analytical model include traditional lumped parameters and In addition, C, D, E, F pm0 are also functions of θ e , and they are also the parameters necessary for establishing the analytical model. They are also the lumped parameters of the analytical model. According to their definitions, it can be seen that C and D are only related to the structure and magnetic saturation of the permanent magnet synchronous motor, while E is related to the structure and magnetic saturation of the permanent magnet synchronous motor, and is also related to the permanent magnet air-gap magnetic density B pm .

[0097] Furthermore, after the step (e), the following may also be included:

[0098] Step (f): Establish a lumped parameter extraction strategy based on finite element method to extract the lumped parameters required for the analytical model.

[0099] The lumped parameters are very important for the analytical model. Affected by magnetic saturation (or current excitation), the lumped parameters under different current excitations are different. In the present invention, for the proposed zero-order radial electromagnetic force analytical model, a set of lumped parameter extraction strategies based on finite element analysis is established, and its specific process is as Figure 6 shown.

[0100] The lumped parameter can be obtained from (I) and (II) through finite element static magnetic field analysis. The stator leakage inductance can be obtained by the method of freezing the magnetic permeability. Specifically, during implementation, by applying unit current excitations i a1u = [1 0 0 0 0 0] T , i a2u = [0 1 0 0 0 0] T , i b1u = [0 0 1 0 0 0]T , i b2u = [0 0 0 1 00] T , i c1u = [0 0 0 0 1 0] T , i c2u = [0 0 0 0 0 1] T , by finite element analysis, integrating the obtained air-gap magnetic flux density, the air-gap magnetic flux linkage vectors generated by each winding can be obtained. For example, when the current excitation is i a1u , the air-gap magnetic flux linkage vector is where B a1u is the air-gap magnetic flux density when the current excitation is i a11u . The stator leakage inductance can be calculated by the following formula.

[0101]

[0102] B pm can also be extracted by the method of freezing the magnetic permeability in finite element analysis. After obtaining B pm , C, D, and E can be obtained through (VIII) and (XIII). In this analysis example, the lumped parameters C, D, E, and F pm0 versus θ e are shown in Figures 7 and 8. Among them, Figure 7 shows C, D, E, and F pm0 when the fundamental current excitation is (30A, 107.7°), and Figure 8 shows C, D, E, and F pm0 when the fundamental current excitation is (60A, 115.2°).

[0103] When only considering the fundamental phase current (i.e., sinusoidal excitation), F r0 obtained by the analytical model and finite element analysis under different fundamental excitations is Figure 9 shown as follows, where the fundamental excitation currents are (15A, 100.8°), (30A, 107.7°), (45A, 111.6°), and (60A, 115.2°) respectively.

[0104] To further verify the accuracy of the analytical model in the present invention, current harmonics are injected on the basis of the fundamental current (i.e., harmonic excitation). Under different harmonic excitations, F r0 obtained by the analytical model and finite element analysis is Figure 9 shown as follows. In the analysis case of the present invention, the orders of the injected current harmonics are 11th and 13th. Among them, the current expression of phase a1 containing harmonics is shown in the following formula.

[0105] i a1 = i 1cos(θ e +α 1 )+i 11 cos(11θ e +α 11 )+i 13 cos(13θ e +α 13 ) (XVII)

[0106] wherein, i 11 and i 13 are the amplitudes of the 11th and 13th current harmonics; α 11 and α 13 are the phases of the 11th and 13th current harmonics.

[0107] The amplitudes and phases of the 11th and 13th current harmonics injected under different fundamental current excitations are shown in Table 2. Analyzing Figure 9 the results shown, it can be seen that except for a certain deviation in the DC component, the fluctuation components in the analytical model results of the present invention have a high degree of fit with the fluctuation components in the finite element analysis results of the prior art, and the fluctuation components in the zero-order force are the main factors causing zero-order vibration. Therefore, this analytical model has high accuracy.

[0108] Table 2 Amplitudes and Phases of the 11th and 13th Current Harmonics Injected under Different Fundamental Currents

[0109]

[0110] The zero-order radial electromagnetic force is the main cause of the stator vibration of the integral-slot-winding interior permanent magnet synchronous motor. The present invention takes the magnetic flux linkage as the key correlation variable, and by reasonably selecting the lumped parameters, reveals the relationship between the air-gap magnetic density distribution and the current. The relationship between the current and the magnetic flux linkage is established using the incremental inductance, the permanent magnet magnetic flux linkage and the armature magnetic flux linkage are separated using the apparent inductance, and the influence of the leakage flux on the air-gap armature magnetic flux linkage is analyzed using the stator leakage inductance. The basic waveform of the air-gap magnetic density distribution is established according to the winding structure, and the characteristics of the armature air-gap magnetic density distribution are further refined by analyzing the influence of the stator structure and the magnetic bridge. In addition to the traditional lumped parameters, new lumped parameters are defined to establish the analytical model. The present invention further establishes a strategy for extracting lumped parameters based on finite element analysis. The analytical model of the present invention has high accuracy compared with the finite element model of the prior art, and can intuitively express the relationship between the current, the lumped parameters and the zero-order radial electromagnetic force.

[0111] The above is the preferred embodiment of the present invention. It should be noted that for those of ordinary skill in the art, without departing from the principle of the present invention, several improvements and refinements can be made, and these improvements and refinements should also be regarded as the protection scope of the present invention.

Claims

1. A zero-order radial force analytical modeling method for a permanent magnet synchronous motor based on lumped parameters, which is used for an integer slot winding interior permanent magnet synchronous motor, characterized in that: include: Step (a): Select flux as the key correlation variable to analyze the relationship between air gap flux distribution and current. The flux type is divided into winding flux and air gap flux according to the observation position, and is divided into armature flux and permanent magnet flux according to the cause. Step (b): using incremental inductance to establish the relationship between current and flux when current harmonics are considered, using apparent inductance to separate permanent magnet flux and armature flux, and using stator leakage inductance to analyze the effect of leakage flux on air gap flux; Step (c): Establish the basic waveform of air gap flux density distribution according to the winding structure, consider the influence of stator structure and magnetic bridge, and further refine the air gap flux density distribution characteristics; Step (d): by analyzing the relationship between air gap flux density and air gap flux linkage, the relationship between lumped parameters, current and air gap armature flux density distribution is established; Step (e): Use the frozen permeability method to extract the air gap permanent magnet flux density, define the lumped parameters C, D, and E related to the stator and rotor structure and the air gap permanent magnet flux density distribution, and establish a zero-order radial electromagnetic force analytical model with explicit current.

2. The method according to claim 1, characterized in that The step (b) comprises: Taking all phase windings of the motor into consideration, the flux linkage relationship is established: Among them, the total magnetic flux ψ=[ψ a1 ψ a2 ψ b1 ψ b2 ψ c1 ψ c2 ] T , a1, a2, b1, b2, c1, c2 are the numbers of the motor phase windings, ψ a1 to c2 is the flux linkage of each phase winding; is the virtual armature flux linkage in the case of incremental inductance; is the virtual permanent magnet flux linkage in the case of incremental inductance; L inc is the incremental inductance matrix; phase current i=[i a1 i a2 i b1 i b2 i c1 i c2 ] T ; a1 phase current i a1 =i1 cos(θ e +α1),θ e =n p θ r is the rotor electrical angle, n p is the number of motor pole pairs, θ r is the rotor mechanical angle; i1 and α1 are the fundamental current amplitude and phase; Incremental Inductance Matrix Among them, l a1a1 represents the self-inductance of phase a1 in the case of incremental inductance, l a1a2 represents the mutual inductance between phase a1 and phase a2 in the case of incremental inductance; By using the apparent inductance to separate the armature flux and the permanent magnet flux, we can get: in, is the armature flux linkage under apparent inductance, is the permanent magnet flux linkage under apparent inductance; After considering the stator leakage flux, the part of the armature flux that passes through the air gap, i.e. the air gap armature flux in, represents the air gap flux linkage of phase a1; is the stator leakage inductance; After considering the stator leakage flux, the air gap armature flux can be obtained according to (I) and (II) as shown in the following formula:

3. The method according to claim 2, characterized in that In the step (c), the total air gap magnetic flux density B g The air gap armature magnetic flux B am and air gap permanent magnet flux density B pm Composition; B am and B pm At the rotor mechanical angle θ r When the spatial angle The spatial distribution relationships of the changes are When the influence of the stator slot structure and the rotor structure is not considered, the magnetic flux density generated by each winding is distributed in the form of a square wave along the air gap. If all windings are considered, the air gap armature flux density is distributed in a step wave along the air gap, which can be expressed as 4. The method according to claim 3, characterized in that The step (c) comprises: Define the stator distribution coefficient m s and rotor distribution coefficient m r .

5. The method according to claim 4, characterized in that The step (c) further comprises: In order to analyze the distribution of air gap magnetic flux density at the magnetic bridge, two main magnetic circuit channels are selected for analysis. The magnetic resistance at the magnetic bridge and the air gap in the magnetic circuit channel are R b and R g , calculated by the following formula: Where, μ0 is the vacuum permeability; μ rb is the relative magnetic permeability of the magnetic bridge; l g is the air gap length; l b is the length of the magnetic bridge; l m is the axial length of the motor; w p is the width of the magnetic circuit channel; The magnetic resistance in different magnetic circuit channels is different, which leads to m at different spatial angles. r The values ​​are different; assuming that at the magnetic bridge, m r Along l b Continuous linear change, m rh and m rl Respectively represent m r The larger and smaller values ​​of , the relationship between them satisfies the following formula: in, is the space electrical angle; m r Need to meet 6. The method according to claim 5, characterized in that The step (d) comprises: Consider the influence coefficient of stator and rotor m s and m r Finally, taking the air gap flux of phase a1 as an example, the flux is related to the integral of the magnetic flux distribution, and the following formula is derived: Where, N is the number of winding turns; r is the air gap radius; arrive is the spatial angle corresponding to the nth tooth on the stator; According to the symmetry of magnetic flux density distribution, (VI) is simplified to the following formula: Considering The step distribution characteristics of (VII) are simplified to the following formula after being expanded by angle: in, n=1...5;B t1 …B t5 Air gap armature flux density The amplitude of each section of the step wave; Define the air gap armature flux vector Taking into account the six-phase winding situation, the following formula is obtained: Among them, A c is the winding coefficient matrix, (IX) The relationship between lumped parameters, current and air-gap armature flux distribution is established.

7. The method according to claim 6, characterized in that The step (e) comprises: Air gap magnetic density B g =B am +B pm , according to Maxwell's tensor equation, the radial electromagnetic force density f r As shown below: Where μ0 is the vacuum magnetic permeability, for f r Integrate along the spatial angle to get the zero-order radial electromagnetic force density F r0 , as shown below: (XI) The zero-order radial electromagnetic force density F r0 It consists of 3 parts, of which the definition is made by B alone. pm The generated part is F pm0 , which can be obtained by freezing the magnetic permeability pm Directly calculated, as shown below: Therefore, F r0 The expression (XI) is converted into the following formula: in, Further simplifying (XIII), we get: Where, D = diag([d1 d2 a3 d4 d5 d6]); E = diag([e1 e2 e3 e4 e5 e6]); Substituting (IX) into (XIV), we can obtain the analytical expression between the zero-order radial electromagnetic force density and the lumped parameter, as shown below: in, (XV) is the established zero-order radial electromagnetic force analytical model.

8. The method according to claim 7, characterized in that The step (e) further comprises: Step (f): Establish a lumped parameter extraction strategy based on finite elements to extract the lumped parameters required for the analytical model.

9. The method according to claim 8, characterized in that The step (f) comprises: Lumped parameters and Through finite element static magnetic field analysis, it is obtained from (I) and (II); and / or, stator leakage inductance Obtained by freezing the permeability method; and / or, B pm Extraction through finite element frozen permeability method; and / or, upon obtaining B pm Thereafter, C, D and E are obtained via (VIII) and (XIII).

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