CN109713972B - Permanent magnet synchronous motor flux linkage optimization method based on nonlinear dynamics analysis - Google Patents

Permanent magnet synchronous motor flux linkage optimization method based on nonlinear dynamics analysis Download PDF

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CN109713972B
CN109713972B CN201811504369.0A CN201811504369A CN109713972B CN 109713972 B CN109713972 B CN 109713972B CN 201811504369 A CN201811504369 A CN 201811504369A CN 109713972 B CN109713972 B CN 109713972B
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permanent magnet
synchronous motor
magnet synchronous
flux linkage
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CN109713972A (en
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解骞
慕长辉
吴刚
郁亚南
贾嵘
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Xian University of Technology
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Abstract

The invention discloses a permanent magnet synchronous motor flux linkage optimization method based on nonlinear dynamics analysis, which comprises the steps of firstly, establishing a mathematical model of a permanent magnet synchronous motor, converting the mathematical model of the permanent magnet synchronous motor by adopting an affine transformation method, then simulating, and determining the flux linkage range of the permanent magnet synchronous motor, wherein the flux linkage range of the permanent magnet synchronous motor can be obtained as the mathematical model of the permanent magnet synchronous motor is stable at a balance point after simulation, and then the Lyapunov index spectrum of the mathematical model of the permanent magnet synchronous motor after simulation is obtained by utilizing matlab simulation software, so that the flux linkage range of the permanent magnet synchronous motor can be further reduced; and finally, building a mathematical model of the simulated permanent magnet synchronous motor, selecting at least 5 flux linkage values for simulation, and obtaining the optimal flux linkage value of the permanent magnet synchronous motor when the flux density is close to saturation. When the method is used for designing the motor, the flux linkage value at the optimal working point can be obtained, and the design cost is reduced.

Description

Permanent magnet synchronous motor flux linkage optimization method based on nonlinear dynamics analysis
Technical Field
The invention belongs to the technical field of nonlinear dynamics, and particularly relates to a permanent magnet synchronous motor flux linkage optimization method based on nonlinear dynamics analysis.
Background
Compared with other motors, the permanent magnet synchronous motor has the advantages of greatly improved performance, small size, light weight, high power factor, fast response, high efficiency, large output torque and the like. Therefore, permanent magnet synchronous motors have higher reliability and have been widely used in industrial and high performance applications, including industrial drives and battery electric vehicles. In recent years, the research on the stability and reliability of the permanent magnet synchronous motor has attracted much attention, because the stable and reliable operation of the permanent magnet synchronous motor is an important issue in industrial automation production. The existing research shows that under certain parameters and working conditions, the permanent magnet synchronous motor presents complex dynamic characteristics. The presence of some special dynamic behavior will seriously affect the stability of the operation of the permanent magnet synchronous motor. Which is mainly manifested by intermittent oscillations in torque and speed, instability in control performance, irregular current noise of the system, and abnormal electromechanical oscillations, which will cause great damage to the motor. However, this phenomenon cannot be suppressed or eliminated by means of conventional linear control methods. As the current field of research has expanded from linear steady-state systems to time-varying systems and non-linear systems, stability analysis has become more and more complex. If the linear analysis theory is continued to be used for researching the problems, not only the precision is poor, but also the essential characteristics are eliminated. Therefore, nonlinear analytical theory is becoming more and more important as a fundamental theory of modern scientific technology and engineering. Therefore, it is necessary to beneficially investigate the mechanism of the permanent magnet synchronous motor by utilizing the nonlinear dynamic behavior.
The permanent magnet synchronous motor is a synchronous motor which generates a synchronous rotating magnetic field by utilizing permanent magnet excitation, and the performance of the permanent magnet synchronous motor is influenced by the permanent magnet. Furthermore, the choice of permanent magnets will affect the cost of permanent magnet synchronous motor design. On the premise of realizing electrical performance indexes, it is very important to scientifically and reasonably design the size of the permanent magnet so as to reduce the manufacturing cost and improve the utilization rate. Permanent magnets are the key of permanent magnet synchronous motors. Some scholars design the permanent magnet by using an intelligent optimization algorithm, so that the quality of the overall design is improved, but the defects exist at the same time, and the determination of a specific objective function is very complex. The nonlinear study of flux linkage optimization of the permanent magnet synchronous motor is rare.
Disclosure of Invention
The invention aims to provide a permanent magnet synchronous motor flux linkage optimization method based on nonlinear dynamics analysis, which can select a proper flux linkage value on the basis of ensuring the stability of a permanent magnet synchronous motor.
The technical scheme adopted by the invention is that a permanent magnet synchronous motor flux linkage optimization method based on nonlinear dynamics analysis is implemented according to the following steps:
step 1, using a d-p coordinate system for analysis, namely synchronously rotating a stator and a rotor, and establishing a mathematical model of the permanent magnet synchronous motor, as shown in formula (1):
Figure RE-GDA0001995255180000021
in the formula (1), LqQ-axis stator inductance; r is stator winding resistance; i.e. iqIs the q-axis stator current; n ispIs the number of pole pairs; l isdIs d-axis stator inductance, and Lq＝Ld(ii) a Omega is the angular speed of the rotor, and the unit is rad/s; i.e. idIs the d-axis stator current; psifIs a permanent magnet flux linkage with the unit of Wb; u. ofqIs the q-axis stator voltage; u. ofdIs the d-axis stator voltage; j is moment of inertia in kg.m2(ii) a b is damping coefficient with unit of N.m. (rad/s)-1；TLIs torque, in units of N · m;
step 2, after the step 1, converting the mathematical model of the permanent magnet synchronous motor by adopting an affine transformation method, wherein the formula is shown as the formula (6):
Figure RE-GDA0001995255180000031
in the formula (6), the reaction mixture is,
Figure RE-GDA0001995255180000032
step 3, simulating the converted mathematical model of the permanent magnet synchronous motor;
step 4, determining the flux linkage range of the permanent magnet synchronous motor, specifically:
step 4.1, solving a mathematical model of the simulated permanent magnet synchronous motor to obtain a balance point of the simulated permanent magnet synchronous motor, as shown in a formula (13);
Figure RE-GDA0001995255180000033
step 4.2, after the step 4.1, obtaining a corresponding Jacobian matrix at the balance point and a characteristic equation at the balance point respectively according to the balance point, wherein the Jacobian matrix and the characteristic equation are shown as a formula (14) and a formula (15);
Figure RE-GDA0001995255180000034
Figure RE-GDA0001995255180000041
step 4.2, after the step 4.1, because of the permanent magnetism after the emulationMathematical model of synchronous motor at balance point P1And P2The position is stable, and according to the Router-Hurwitz criterion, the mathematical model of the simulated permanent magnet synchronous motor needs to satisfy the formula (16);
Figure RE-GDA0001995255180000042
solving the formula (16) to obtain a solution x;
if 0<Theta is less than or equal to x, the flux linkage range of the permanent magnet synchronous motor obtained by the formula (16) is
Figure RE-GDA0001995255180000043
If theta>x, the flux linkage range of the permanent magnet synchronous motor obtained by the formula (16) is
Figure RE-GDA0001995255180000044
Step 5, after the step 4, utilizing matlab simulation software to obtain a Lyapunov index spectrum of the simulated mathematical model of the permanent magnet synchronous motor, and knowing three Lyapunov indexes lambda of the simulated mathematical model of the permanent magnet synchronous motor according to the Lyapunov stability theorem1、λ2And λ3When the flux linkage ranges are all less than 0, the flux linkage ranges are stable, and the flux linkage ranges of the permanent magnet synchronous motor can be further reduced;
and 6, building a mathematical model of the simulated permanent magnet synchronous motor by using electromagnetic field analysis software, selecting at least 5 flux linkage values from the flux linkage range of the permanent magnet synchronous motor obtained in the step 5 for simulation to obtain flux density, and obtaining the optimal flux linkage value of the permanent magnet synchronous motor when the flux density is close to saturation.
The present invention is also characterized in that,
in the step 1, corresponding equations can be obtained by a mathematical model of the permanent magnet synchronous motor, wherein the equations comprise a voltage equation, a flux linkage equation, a torque equation and a motion equation;
the voltage equation is shown in equation (2):
Figure RE-GDA0001995255180000051
in the formula (2), ωeIs the electrical angular frequency; psiqIs a q-axis stator flux linkage; psidIs a d-axis stator flux linkage;
the flux linkage equation is shown in equation (3):
Figure RE-GDA0001995255180000052
the torque equation is shown in equation (4):
Figure RE-GDA0001995255180000053
in the formula (4), TeIs the electromagnetic torque with the unit of N.m;
the equation of motion, as shown in equation (5):
Figure RE-GDA0001995255180000054
in the formula (5), B is a magnetic flux density.
Step 2, specifically: let the affine transformation form be:
Figure RE-GDA0001995255180000055
thus, can obtain
Figure RE-GDA0001995255180000056
Let another affine transformation form be
Figure RE-GDA0001995255180000057
The three-dimensional power system is
Figure RE-GDA0001995255180000061
And the balance point and the stability of the mathematical model of the permanent magnet synchronous motor are kept;
the definitions Λ, M and N are shown as formulas (7), (8) and (9), respectively:
Figure RE-GDA0001995255180000062
Figure RE-GDA0001995255180000063
Figure RE-GDA0001995255180000064
substituting equations (7), (8) and (9) into affine transformation form
Figure RE-GDA0001995255180000065
To obtain formula (10) and formula (11);
Figure RE-GDA0001995255180000066
Figure RE-GDA0001995255180000071
based on
Figure RE-GDA0001995255180000072
And a three-dimensional power system
Figure RE-GDA0001995255180000073
The transformed mathematical model of the permanent magnet synchronous motor can be obtained.
The step 3 specifically comprises the following steps:
step 3.1, designing a primary model of the permanent magnet synchronous motor by adopting ANSYS software, and ordering
Figure RE-GDA0001995255180000074
Converting the formula (6) into a simulated mathematical model of the permanent magnet synchronous motor, as shown in a formula (12);
Figure RE-GDA0001995255180000075
step 3.2, obtaining relevant parameters of the permanent magnet synchronous motor according to the primary model of the permanent magnet synchronous motor: resistance R and magnetic logarithm n of stator windingPDamping coefficient b and moment of inertia J, from the formula
Figure RE-GDA0001995255180000076
And
Figure RE-GDA0001995255180000077
mu and theta are calculated.
The invention has the beneficial effects that:
when the method is used for designing the motor, the flux linkage value at the optimal working point can be obtained, the design cost is reduced, and the utilization rate of the permanent magnet synchronous motor is improved as much as possible on the premise of ensuring the stability of the motor to be kept.
Drawings
FIG. 1 is a 2D model of a permanent magnet synchronous machine in an embodiment of the invention;
FIG. 2 is a 3D model of a permanent magnet synchronous machine in an embodiment of the invention;
FIG. 3 is a Lyapunov exponent spectra in an embodiment of the present invention;
FIG. 4 is a kinetic bifurcation diagram in an embodiment of the present invention;
FIG. 5 is a cloud of magnetic flux densities in an embodiment of the invention;
FIG. 6 is a diagram of flux linkage waveforms in an embodiment of the present invention.
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.
The invention relates to a permanent magnet synchronous motor flux linkage optimization method based on nonlinear dynamics analysis, which is implemented according to the following steps:
step 1, using a d-p coordinate system for analysis, namely synchronously rotating a stator and a rotor, and establishing a mathematical model of the permanent magnet synchronous motor, as shown in formula (1):
Figure RE-GDA0001995255180000081
in the formula (1), LqQ-axis stator inductance; r is stator winding resistance; i.e. iqIs the q-axis stator current; n ispIs the number of pole pairs; l isdIs d-axis stator inductance, and Lq＝Ld(ii) a Omega is the angular speed of the rotor, and the unit is rad/s; i.e. idIs the d-axis stator current; psifIs a permanent magnet flux linkage with the unit of Wb; u. ofqIs the q-axis stator voltage; u. ofdIs the d-axis stator voltage; j is moment of inertia in kg.m2(ii) a b is damping coefficient with unit of N.m. (rad/s)-1；TLIs torque, in units of N · m;
corresponding equations can be obtained through a mathematical model of the permanent magnet synchronous motor, and the equations comprise a voltage equation, a flux linkage equation, a torque equation and a motion equation;
the voltage equation is shown in equation (2):
Figure RE-GDA0001995255180000091
in the formula (2), ωeIs the electrical angular frequency; psiqIs a q-axis stator flux linkage; psidIs a d-axis stator flux linkage;
the flux linkage equation is shown in equation (3):
Figure RE-GDA0001995255180000092
the torque equation is shown in equation (4):
Figure RE-GDA0001995255180000093
in the formula (4), TeIs the electromagnetic torque with the unit of N.m;
the equation of motion, as shown in equation (5):
Figure RE-GDA0001995255180000094
in the formula (5), B is a magnetic flux density;
step 2, after the step 1, converting the mathematical model of the permanent magnet synchronous motor by adopting an affine transformation method, wherein the formula is shown as the formula (6):
Figure RE-GDA0001995255180000095
in the formula (6), the reaction mixture is,
Figure RE-GDA0001995255180000101
the method specifically comprises the following steps: let the affine transformation form be:
Figure RE-GDA0001995255180000102
thus, can obtain
Figure RE-GDA0001995255180000103
Let another affine transformation form be
Figure RE-GDA0001995255180000104
The three-dimensional power system is
Figure RE-GDA0001995255180000105
And the balance point and the stability of the mathematical model of the permanent magnet synchronous motor are kept;
the definitions Λ, M and N are shown as formulas (7), (8) and (9), respectively:
Figure RE-GDA0001995255180000106
Figure RE-GDA0001995255180000107
Figure RE-GDA0001995255180000108
substituting equations (7), (8) and (9) into affine transformation form
Figure RE-GDA0001995255180000109
To obtain formula (10) and formula (11);
Figure RE-GDA0001995255180000111
Figure RE-GDA0001995255180000112
based on
Figure RE-GDA0001995255180000113
And a three-dimensional power system
Figure RE-GDA0001995255180000114
A converted mathematical model of the permanent magnet synchronous motor can be obtained;
step 3, simulating the converted mathematical model of the permanent magnet synchronous motor; the method specifically comprises the following steps:
step 3.1, designing a primary model of the permanent magnet synchronous motor by adopting ANSYS software, and ordering
Figure RE-GDA0001995255180000115
Converting the formula (6) into a simulated mathematical model of the permanent magnet synchronous motor, as shown in a formula (12);
Figure RE-GDA0001995255180000116
and 3.2, obtaining relevant parameters of the permanent magnet synchronous motor according to the primary model of the permanent magnet synchronous motor: resistance R and magnetic logarithm n of stator windingPDamping coefficient b and moment of inertia J, from the formula
Figure RE-GDA0001995255180000117
And
Figure RE-GDA0001995255180000118
calculating to obtain mu and theta;
step 4, determining the flux linkage range of the permanent magnet synchronous motor, specifically:
step 4.1, solving a mathematical model of the simulated permanent magnet synchronous motor to obtain a balance point of the simulated permanent magnet synchronous motor, as shown in a formula (13);
Figure RE-GDA0001995255180000121
step 4.2, after the step 4.1, obtaining a corresponding Jacobian matrix at the balance point and a characteristic equation at the balance point respectively according to the balance point, wherein the Jacobian matrix and the characteristic equation are shown as a formula (14) and a formula (15);
Figure RE-GDA0001995255180000122
Figure RE-GDA0001995255180000123
step 4.2, after the step 4.1, because the mathematical model of the simulated permanent magnet synchronous motor is at the balance point P1And P2The position is stable, and according to the Router-Hurwitz criterion, the mathematical model of the simulated permanent magnet synchronous motor needs to satisfy the formula (16);
Figure RE-GDA0001995255180000124
solving the formula (16) to obtain a solution x;
if 0<Theta is less than or equal to x, the flux linkage range of the permanent magnet synchronous motor obtained by the formula (16) is
Figure RE-GDA0001995255180000131
If theta>x, the flux linkage range of the permanent magnet synchronous motor obtained by the formula (16) is
Figure RE-GDA0001995255180000132
Step 5, after the step 4, utilizing matlab simulation software to obtain a Lyapunov index spectrum of the simulated mathematical model of the permanent magnet synchronous motor, and knowing three Lyapunov indexes lambda of the simulated mathematical model of the permanent magnet synchronous motor according to the Lyapunov stability theorem1、λ2And λ3When both are less than 0, it is stable, i.e. one can proceedThe flux linkage range of the permanent magnet synchronous motor is narrowed;
and 6, building a mathematical model of the simulated permanent magnet synchronous motor by using electromagnetic field analysis software ANSYSMAXwell (Version 14.5), selecting at least 5 flux linkage values from the flux linkage range of the permanent magnet synchronous motor obtained in the step 5 for simulation to obtain flux density, and obtaining the optimal flux linkage value of the permanent magnet synchronous motor when the flux density is close to saturation (namely B is approximately equal to 1.6T).
Examples
The invention relates to a permanent magnet synchronous motor flux linkage optimization method based on nonlinear dynamics analysis, which is implemented according to the following steps:
step 1, using a d-p coordinate system for analysis, namely synchronously rotating a stator and a rotor, and establishing a mathematical model of the permanent magnet synchronous motor, as shown in formula (1):
Figure RE-GDA0001995255180000133
in the formula (1), LqQ-axis stator inductance; r is stator winding resistance; i.e. iqIs the q-axis stator current; n ispIs the number of pole pairs; l isdIs d-axis stator inductance, and Lq＝Ld(ii) a Omega is the angular speed of the rotor, and the unit is rad/s; i.e. idIs the d-axis stator current; psifIs a permanent magnet flux linkage with the unit of Wb; u. ofqIs the q-axis stator voltage; u. ofdIs the d-axis stator voltage; j is moment of inertia in kg.m2(ii) a b is damping coefficient with unit of N.m. (rad/s)-1；TLIs torque, in units of N · m;
corresponding equations can be obtained through a mathematical model of the permanent magnet synchronous motor, and the equations comprise a voltage equation, a flux linkage equation, a torque equation and a motion equation;
the voltage equation is shown in equation (2):
Figure RE-GDA0001995255180000141
in the formula (2), ωeIs the electrical angular frequency; psiqIs a q-axis stator flux linkage;ψdis a d-axis stator flux linkage;
the flux linkage equation is shown in equation (3):
Figure RE-GDA0001995255180000142
the torque equation is shown in equation (4):
Figure RE-GDA0001995255180000143
in the formula (4), TeIs the electromagnetic torque with the unit of N.m;
the equation of motion, as shown in equation (5):
Figure RE-GDA0001995255180000144
in the formula (5), B is a magnetic flux density;
step 2, after the step 1, converting the mathematical model of the permanent magnet synchronous motor by adopting an affine transformation method, wherein the formula is shown as the formula (6):
Figure RE-GDA0001995255180000151
in the formula (6), the reaction mixture is,
Figure RE-GDA0001995255180000152
the method specifically comprises the following steps: let the affine transformation form be:
Figure RE-GDA0001995255180000153
thus, can obtain
Figure RE-GDA0001995255180000154
Let another affine transformation form be
Figure RE-GDA0001995255180000155
The three-dimensional power system is
Figure RE-GDA0001995255180000156
And the balance point and the stability of the mathematical model of the permanent magnet synchronous motor are kept;
the definitions Λ, M and N are shown as formulas (7), (8) and (9), respectively:
Figure RE-GDA0001995255180000157
Figure RE-GDA0001995255180000158
Figure RE-GDA0001995255180000159
substituting equations (7), (8) and (9) into affine transformation form
Figure RE-GDA00019952551800001510
To obtain formula (10) and formula (11);
Figure RE-GDA0001995255180000161
Figure RE-GDA0001995255180000162
based on
Figure RE-GDA0001995255180000163
And a three-dimensional power system
Figure RE-GDA0001995255180000164
A converted mathematical model of the permanent magnet synchronous motor can be obtained;
step 3, simulating the converted mathematical model of the permanent magnet synchronous motor; the method specifically comprises the following steps:
an initial model of the 30kW permanent magnet synchronous motor is designed by using ANSYS software, as shown in formula (12), so as to obtain related analysis parameters and rated parametersAnd 30kW PMSM 2D and 3D models, as shown in FIGS. 1 and 2, with the major dimensions listed in tables 1 and 2, expressed by the formula
Figure RE-GDA0001995255180000165
And
Figure RE-GDA0001995255180000166
calculating to obtain mu and theta, wherein mu is 9.85, and theta is 3.97;
Figure RE-GDA0001995255180000167
table 130 kW permanent magnet synchronous motor rated parameter and size
Figure RE-GDA0001995255180000168
Figure RE-GDA0001995255180000171
TABLE 2 permanent magnet synchronous motor Circuit parameters
Parameter(s) Numerical value
R
5×10-2Ω
Lq 3.6×10-5H
Ld 3.6×10-5H
nP 4
b 2.03×10-2N·m·s/rad
J 8.06×10-2kg·m2
Step 4, determining the flux linkage range of the permanent magnet synchronous motor, specifically:
step 4.1, solving a mathematical model of the simulated permanent magnet synchronous motor to obtain a balance point of the simulated permanent magnet synchronous motor, as shown in a formula (13);
Figure RE-GDA0001995255180000172
step 4.2, after the step 4.1, obtaining a corresponding Jacobian matrix at the balance point and a characteristic equation at the balance point respectively according to the balance point, wherein the Jacobian matrix and the characteristic equation are shown as a formula (14) and a formula (15);
Figure RE-GDA0001995255180000173
Figure RE-GDA0001995255180000181
step 4.2, after the step 4.1, because the mathematical model of the simulated permanent magnet synchronous motor is at the balance point P1And P2The position is stable, and according to the Router-Hurwitz criterion, the mathematical model of the simulated permanent magnet synchronous motor needs to satisfy the formula (16);
Figure RE-GDA0001995255180000182
solving the formula (16) to obtain a solution of 3; if 0<Theta is less than or equal to 3, then
Figure RE-GDA0001995255180000183
Thus, ψf> 0.0829; if θ > 3, then
Figure RE-GDA0001995255180000184
Thus, 0.0829<ψf<0.4323；
Step 5, after the step 4, obtaining a Lyapunov index spectrum of the simulated mathematical model of the permanent magnet synchronous motor by utilizing matlab simulation software, as shown in FIG. 3, according to the Lyapunov stability theorem, knowing three Lyapunov indexes lambda of the simulated mathematical model of the permanent magnet synchronous motor1、λ2And λ3When the values are less than 0, the magnetic flux linkage stability range of the permanent magnet synchronous motor is stable, and therefore, according to the lyapunov exponent spectrum in fig. 3, the following conclusion can be drawn: when 0 is present<ψf<0.334, when the system keeps a stable motion state; when 0.334<ψfAnd when the system is in a complex dynamic behavior, the system is in an unstable state. And analyzing the dynamic characteristics of the nonlinear system by using bifurcation mapping under the condition of changing system parameters. Bifurcation is the main path leading to chaos in a stable state, the bifurcation diagram changes as shown in FIG. 4, obviously, the behavior of the system dynamics bifurcation diagram is completely the same as that of the Lyapunov exponential spectrum, when 0<ψf<0.334, when the system maintains a stable motion state, the system begins to become unstable as soon as this interval is exceeded, and at the same time, with ψfA complex dynamic behavior phenomenon occurs. In order to ensure that the motor stably runs in the optimal working state, the optimal value of the flux linkage needs to be searched in the range;
step 6, building a mathematical model of the simulated permanent magnet synchronous motor by using electromagnetic field analysis software ANSYSMAXwell (Version 14.5), selecting at least 5 flux linkage values from the flux linkage range of the permanent magnet synchronous motor obtained in the step 5 for simulation to obtain flux density, and obtaining the optimal flux linkage value of the permanent magnet synchronous motor when the flux density is close to saturation (namely B is approximately equal to 1.6T); FIGS. 5 and 6 show the equation psifFlux density at 0.3160WbThe cloud pattern and the waveform of the magnetic flux in this case, B ═ 1.6T is obtained, which means that the magnetic flux density is close to saturation. Thus, the saturation point of the flux density of each stator tooth and stator yoke is discontinuous, i.e. the magnetic work of the permanent magnet at the optimum operating point, which means that the machine material will be maximized and the performance will be optimized.
Corresponding permanent magnet size and dimensions in the optimum state, i.e. psif0.3160 Wb. As shown in table 3, the corresponding rotor data and stator data are shown in tables 4 and 5, and table 6 is the winding arrangement.
TABLE 3 permanent magnet Specifications and dimensions
Figure RE-GDA0001995255180000191
Figure RE-GDA0001995255180000201
TABLE 4 rotor data
Parameter(s) Numerical value
Minimum air gap 1.55mm
Inner diameter 63mm
Rotor length 310mm
Core factor 0.95
Core type 50W600
Radius of arc 67.15mm
Spatial extent of electrodynamic polar arc 0.9
Electrode plate 0.899194
Maximum thickness of magnet 6.6mm
Width of magnet 47.3268mm
Magnet type N35SH
Rotor type
2
TABLE 5 stator data
Parameter(s) Numerical value
Number of stator slots 36
Outer diameter of stator 247.5mm
Stator bore 140mm
Stator slot type 3
Width of top tooth 6.44mm
Width of bottom tooth 6.44mm
Stator core length 310mm
Stator core stacking factor 0.95
TABLE 6 winding arrangement
Figure RE-GDA0001995255180000202
Figure RE-GDA0001995255180000211

Claims (3)

1. A permanent magnet synchronous motor flux linkage optimization method based on nonlinear dynamics analysis is characterized by comprising the following steps:
step 1, using a d-p coordinate system for analysis, namely synchronously rotating a stator and a rotor, and establishing a mathematical model of the permanent magnet synchronous motor, as shown in formula (1):
Figure FDA0002468321480000011
in the formula (1), LqQ-axis stator inductance; r is stator winding resistance; i.e. iqIs the q-axis stator current; n ispIs the number of pole pairs; l isdIs d-axis stator inductance, and Lq＝Ld(ii) a Omega is the angular speed of the rotor, and the unit is rad/s; i.e. idIs the d-axis stator current; psifIs a permanent magnet flux linkage with the unit of Wb; u. ofqIs the q-axis stator voltage; u. ofdIs the d-axis stator voltage; j is moment of inertia in kg.m2(ii) a b is damping coefficient with unit of N.m. (rad/s)-1；TLIs torque, in units of N · m;
step 2, after the step 1, converting the mathematical model of the permanent magnet synchronous motor by adopting an affine transformation method, wherein the formula is shown as the formula (6):
Figure FDA0002468321480000012
in the formula (6), the reaction mixture is,
Figure FDA0002468321480000013
the method specifically comprises the following steps: let the affine transformation form be:
Figure FDA0002468321480000014
thus, can obtain
Figure FDA0002468321480000015
Let another affine transformation form be
Figure FDA0002468321480000021
The three-dimensional power system is
Figure FDA0002468321480000022
And the balance point and the stability of the mathematical model of the permanent magnet synchronous motor are kept;
the definitions Λ, M and N are shown as formulas (7), (8) and (9), respectively:
Figure FDA0002468321480000023
Figure FDA0002468321480000024
Figure FDA0002468321480000025
substituting equations (7), (8) and (9) into affine transformation form
Figure FDA0002468321480000026
To obtain formula (10) and formula (11);
Figure FDA0002468321480000027
Figure FDA0002468321480000031
based on
Figure FDA0002468321480000032
And a three-dimensional power system
Figure FDA0002468321480000033
A converted mathematical model of the permanent magnet synchronous motor can be obtained;
step 3, simulating the converted mathematical model of the permanent magnet synchronous motor;
step 4, determining the flux linkage range of the permanent magnet synchronous motor, specifically:
step 4.1, solving a mathematical model of the simulated permanent magnet synchronous motor to obtain a balance point of the simulated permanent magnet synchronous motor, as shown in a formula (13);
Figure FDA0002468321480000034
step 4.2, after the step 4.1, obtaining a corresponding Jacobian matrix at the balance point and a characteristic equation at the balance point respectively according to the balance point, wherein the Jacobian matrix and the characteristic equation are shown as a formula (14) and a formula (15);
Figure FDA0002468321480000035
Figure FDA0002468321480000036
step 4.2, after the step 4.1, because the mathematical model of the simulated permanent magnet synchronous motor is at the balance point P1And P2The position is stable, and according to the Router-Hurwitz criterion, the mathematical model of the simulated permanent magnet synchronous motor needs to satisfy the formula (16);
Figure FDA0002468321480000041
solving the formula (16) to obtain a solution x;
if 0<Theta is less than or equal to x, the flux linkage range of the permanent magnet synchronous motor obtained by the formula (16) is
Figure FDA0002468321480000042
If theta>x, the flux linkage range of the permanent magnet synchronous motor obtained by the formula (16) is
Figure FDA0002468321480000043
Step 5, after the step 4, utilizing matlab simulation software to obtain a Lyapunov index spectrum of the simulated mathematical model of the permanent magnet synchronous motor, and knowing three Lyapunov indexes lambda of the simulated mathematical model of the permanent magnet synchronous motor according to the Lyapunov stability theorem1、λ2And λ3When the flux linkage ranges are all less than 0, the flux linkage ranges are stable, and the flux linkage ranges of the permanent magnet synchronous motor can be further reduced;
and 6, building a mathematical model of the simulated permanent magnet synchronous motor by using electromagnetic field analysis software, selecting at least 5 flux linkage values from the flux linkage range of the permanent magnet synchronous motor obtained in the step 5 for simulation to obtain flux density, and obtaining the optimal flux linkage value of the permanent magnet synchronous motor when the flux density is close to saturation.
2. The method for optimizing the flux linkage of the permanent magnet synchronous motor based on the nonlinear dynamical analysis of the claim 1, wherein in the step 1, corresponding equations including a voltage equation, a flux linkage equation, a torque equation and a motion equation can be obtained from a mathematical model of the permanent magnet synchronous motor;
the voltage equation is shown in equation (2):
Figure FDA0002468321480000051
in the formula (2), ωeIs the electrical angular frequency; psiqIs a q-axis stator flux linkage; psidIs a d-axis stator flux linkage;
the flux linkage equation is shown in equation (3):
Figure FDA0002468321480000052
the torque equation is shown in equation (4):
Figure FDA0002468321480000053
in the formula (4), TeIs the electromagnetic torque with the unit of N.m;
the equation of motion, as shown in equation (5):
Figure FDA0002468321480000054
in the formula (5), B is a magnetic flux density.
3. The method for optimizing the flux linkage of the permanent magnet synchronous motor based on the nonlinear dynamical analysis according to claim 1, wherein the step 3 specifically comprises:
step 3.1, designing a primary model of the permanent magnet synchronous motor by adopting ANSYS software, and ordering
Figure FDA0002468321480000055
Converting the formula (6) into a simulated mathematical model of the permanent magnet synchronous motor, as shown in a formula (12);
Figure FDA0002468321480000061
and 3.2, obtaining relevant parameters of the permanent magnet synchronous motor according to the primary model of the permanent magnet synchronous motor: resistance R and magnetic logarithm n of stator windingPDamping coefficient b and moment of inertia J, from the formula
Figure FDA0002468321480000062
And
Figure FDA0002468321480000063
mu and theta are calculated.
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