CN110955960A - Permanent magnet spherical motor tooth socket torque analysis method based on equivalent magnetic circuit method - Google Patents
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Abstract
The invention relates to a permanent magnet spherical motor tooth socket torque analysis method based on an equivalent magnetic circuit method, which comprises the following steps: calculating the air gap flux density generated by a single flux tube; calculating the cogging torque of the flux tube on the stator teeth; cogging torque produced by a single pole pair stator tooth: calculating the cogging torque generated by the flux tube according to the magnetic flux tube spherical coordinates, and performing vector superposition on the cogging torque to obtain the cogging torque generated by the single-pole stator teeth; cogging torque produced by all pole pairs stator teeth: and calculating the cogging torque of each magnetic pole on the stator teeth according to the initial position of each magnetic pole in the permanent magnet spherical motor, and then performing vector superposition on the cogging torque to obtain the cogging torque of the whole stator structure.
Description
Technical Field
The invention belongs to the field of analysis of cogging torque of spherical motors, and particularly relates to an analysis method of cogging torque of an iron shell and tooth spherical motor.
Background
The spherical motor has the advantages of small volume, high response speed, high motion precision, direct drive, nonsingular working space, high efficiency and the like, so that the spherical motor becomes an effective solution for eliminating the defects of the multi-degree-of-freedom device; and the spherical motor has wide application prospect in the fields of aerospace, intelligent robots, automobile manufacturing, precision equipment, medical operations and the like. With the development of high-performance rare earth materials, researchers introduce permanent magnets into the spherical motor, so that the size of the spherical motor is greatly reduced, and the magnetic energy product, the operation efficiency and the controllability of the ball machine are improved. Therefore, the permanent magnet spherical motor has become the subject of extensive research by domestic and foreign scholars for decades.
At present, in order to reduce the complexity of analysis, a stator yoke, a stator tooth and a rotor yoke of a domestic and international permanent magnet type spherical motor are usually made of non-ferromagnetic materials, and under the condition, the air gap magnetic flux density and the electromagnetic torque are small, so that the energy utilization rate of the spherical motor is low, and when the three are made of ferromagnetic materials, a magnetic circuit changes along with the change of the position of a rotor, so that the analysis of the cogging torque becomes more complex. In the research, aiming at increasing the electromagnetic torque of the permanent magnet type spherical motor, a stator yoke, a rotor yoke and stator teeth of the spherical motor are made of ferromagnetic materials. Due to the complexity of the spherical motor structure, the analysis and calculation amount of the cogging torque by adopting a finite element method is large and the consumed time is long; and when the structural parameters of the permanent magnet spherical motor are changed, the simulation is carried out again. Therefore, the finite element method is not suitable for use in the early stages of motor design and optimization. In the research, an equivalent magnetic circuit method is selected to quickly and accurately calculate the cogging torque, and the model is applied to the control of the spherical motor, so that the effect of inhibiting the torque pulsation and the noise is hopefully achieved by changing the input current to compensate the electromagnetic torque.
Disclosure of Invention
The invention aims to overcome the defect that the control error is caused by the cogging torque generated when the electromagnetic torque of a spherical motor is increased, and provides a method for quickly analyzing the cogging torque of an iron shell and tooth spherical motor when a rotor is at any position. The invention adopts a discrete numerical analysis method and a method of combining an equivalent magnetic circuit and a side force to analyze the cogging torque of a permanent magnet spherical motor, and the specific scheme is as follows:
a permanent magnet spherical motor tooth socket torque analysis method based on an equivalent magnetic circuit method comprises the following steps:
the first step is as follows: calculating the air gap flux density generated by a single flux tube: according to a single poleThe permanent magnet spherical motor structure is characterized in that a permanent magnet spherical motor tooth-groove torque model is established by adopting a discretization numerical analysis method, a stator winding is not considered, a permanent magnet is selected as a unique magnetomotive force source of the permanent magnet spherical motor, magnetic poles are discretely divided according to spherical coordinates, and an included angle theta 1 of the spherical coordinates occupied by the magnetic poles is divided into N according to the longitude theta directioniIn terms of latitude
Direction divides 2 pi into NjCalculating the initial spherical coordinates of each magnetic flux tube
Comprises the following steps:
wherein r isrIs the radius of the magnetic pole, RrIs the outer diameter of the magnetic pole of the rotor,
as the initial spherical coordinates of the flux tube, (x)0,y0,z0) The initial rectangular coordinate of the magnetic flux tube is shown, subscript i, j is the label of the magnetic flux tube, N+The magnetic flux tube represents a positive integer, magnetic force lines on the permanent magnet are set to be uniformly distributed, and the magnetic flux tube is divided into four conditions according to different inflow positions of the magnetic flux tube: the first is that the magnetic flux tube flows into the side of the stator iron teeth, and the magnetic flux tube generates side force at the moment; the second is that the flux tube flows into the stator yoke part, and the flux tube does not generate side force; the third is that the magnetic flux tube flows into the bottom of the stator tooth, and the magnetic flux tube does not generate side force; the fourth is that the magnetic flux tube flows into the rotor ball, and the magnetic flux tube does not generate side force at the moment; the path lengths of the flux tubes at different positions are obtained according to the four conditions, and the air gap flux density B is obtained on the basis
Where μ is the air gap permeability, w is the flux path length in the air gap, H is the magnetic field strength, lpmIs the magnetic pole height;
the second step is that: calculating the cogging torque of the flux tube on the stator teeth: because the magnetic flux tube flows into the side edge of the stator iron tooth to generate side force and tooth space torque, the tooth space torque under the condition is calculated, firstly, the side force F generated by a single magnetic flux tube is calculated by a side force method as follows:
△ S is the infinitesimal area of the magnetic pole sphere ejected by the flux tube;
secondly, obtaining a force arm R of the magnetic flux tube generating cogging torque according to the position of the magnetic flux tube flowing into the side edge of the stator tooth:
wherein R issDenotes the stator tooth inner diameter, rsThe radius of a stator cylinder is defined, and t is the projection length of an emergent point and an incident point of the flux tube on the spherical surface at the bottom of the stator tooth;
finally, the cogging torque T generated by the flux tube acting on the stator teeth is obtained as F multiplied by R, and the cogging torque is decomposed along a rectangular coordinate system to obtain torque components which are respectively as follows:
wherein Fx, Fy and Fz are components of side force under a rectangular coordinate system respectively, Rx, Ry and Rz are components of force arm under the rectangular coordinate system respectively, and the incident point-sphere coordinate of the magnetic flux tube is
The coordinates of the emergent point and the sphere of the magnetic flux tube are
The stator tooth spherical coordinate is
∠ AOB is the included angle between the outgoing line of the magnetic flux tube and the center line of the stator teeth;
the third step: cogging torque produced by a single pole pair stator tooth: calculating the cogging torque generated by the flux tube according to the magnetic flux tube spherical coordinates, and performing vector superposition on the cogging torque to obtain the cogging torque generated by the single-pole stator teeth;
the fourth step: cogging torque produced by all pole pairs stator teeth: and (2) calculating the cogging torque of each magnetic pole on the stator teeth according to the initial position of each magnetic pole in the permanent magnet spherical motor, and then performing vector superposition on the cogging torque to obtain the cogging torque of the whole stator structure:
wherein, TckCogging torque for the kth pole pair stator teeth, TcogAnd (3) the cogging torque generated by the whole rotor is obtained, n is the number of magnetic poles of the rotor, Euler rotation change is used for expressing the rotation change of the rotor of the spherical motor, and the position of the magnetic poles of the rotor is changed along with the Euler rotation change, so that the cogging torque of the rotor of the spherical motor at any position is obtained.
Drawings
FIG. 1: structural schematic diagram of monopole permanent magnet spherical motor
FIG. 2: (a) unipolar spherical coordinate diagram (b) magnetic pole grid division diagram
FIG. 3: magnetic pole selection tooth plane schematic diagram
FIG. 4: 2-D magnetic line and tooth included angle schematic diagram
FIG. 5: 3-D magnetic line and tooth included angle schematic diagram
FIG. 6: magnetic flux path model schematic
FIG. 7: magnetic leakage model schematic diagram
FIG. 8: schematic structure diagram of multi-pole permanent magnet spherical motor
FIG. 9: cogging torque result chart of unipolar permanent magnet spherical motor
FIG. 10: schematic diagram of motion trail of multi-pole permanent magnet spherical motor
FIG. 11: tooth space torque result graph of multi-pole permanent magnet spherical motor
Detailed Description
The invention provides a method for solving cogging torque based on an equivalent magnetic circuit method for an iron permanent magnet spherical motor, and the method can quickly and accurately calculate the cogging torque of the permanent magnet spherical motor. The invention will be described in detail with reference to the accompanying drawings and simulation examples, and the specific implementation steps are as follows:
(1) flux tube routing
The structure researched by the invention is a permanent magnet spherical motor which contains Ns stator teeth, and the structure can be shown in figure 1. Discretizing the rotor magnetic poles according to the form in figure 2, and dividing theta according to theta direction1Is divided into NiThe preparation method comprises the following steps of (1),
direction divides 2 pi into NjFlux tube perpendicularly projected from magnetic pole surface
Reduced to a magnetic line of force, wherein RrAccording to the distance between the centers of the spheres and the shortest path, the upper surface of the rotor radius magnetic pole is projected to the nearest tooth, and considering that the lengths g of air gaps between the poles are the same, the tooth closest to the exit point of the magnetic line can be found by judging the length of the magnetic line of the arc segment as shown in fig. 3. The magnetic force lines pass through the straight line and the circular arc part to vertically inject into the side edge of the tooth, the injection direction and the tooth center line
(k is 1,2 … Ns) perpendicular to the tooth centerline as shown in fig. 4, where the short flux path is equivalent to a short arc, equivalent to a short straight line GP, and equivalent to a short GQ. FIG. 5 shows the structure of two iron teeth and one magnetic poleAnd the included angle between the magnetic force line and the tooth center line is zeta, the smaller zeta is, and the magnetic force line at the circular arc is shorter.
In FIG. 4, vectors
And
the interval can be represented as:
wherein
Is a vector in the direction of the stator teeth,
is the flux tube direction vector, RrWhich is the radius of the rotor, is,
is the coordinate of the tooth sphere of the stator,
and the coordinates of the magnetic flux tube sphere.
From the cosine theorem, the included angle ζ between two vectors can be obtained:
therefore, the included angle zeta between the magnetic line and each other tooth is calculated in turniAnd find out the minimum zetaminThus obtaining the stator teeth interlinked with the magnetic lines of force.
(2) Magnetic density calculation
As shown in fig. 6, the flux path consists of straight lines and circular arcs, and the thickness g of the air gap layer of the spherical motor is thick. Here, the arc is approximated as a straight line, so the arc radius t is expressed as:
t＝Rsarcsin(∠AOB-∠COB) (8)
wherein ∠ AOB is the angle formed by the magnetic line of force emitting direction and the optimum tooth center line, RsThe stator tooth inner diameter is represented as:
Rs＝Rr+g (9)
where g is the air gap length, ∠ COB is:
wherein r issFor the stator cylinder radius, the expression for the flux path length w in the air gap can be obtained from equations (8) - (10):
wherein lpmIs the pole height.
In particular, there are three special cases at this time according to the magnetic shortest path principle.
A. When the magnetic pole is positioned under the stator tooth, the magnetic force line is ejected from the magnetic pole and directly ejected into the iron tooth from the tooth bottom, no side force and tooth slot torque are generated, and the included angles formed by the ejection direction of the partial magnetic force line and the optimal tooth central line are all smaller than arcsin (rs/Rs) Therefore, the magnetic line determination condition for generating cogging torque is:
the length of the magnetic circuit when the magnetic line of force is injected into the bottom of the tooth is as follows:
w＝lpm+g (13)
B. when the length of the magnetic path is longer than the magnetic flux path (l) of the magnetic line vertically injected into the stator yokepm+g+hs) In which h issSince the magnetic flux is injected into the stator yoke to the stator tooth height and no cogging torque is generated, the magnetic flux determining conditions for generating cogging torque are as follows:
w≤lpm+g+hs(14)
the magnetic path length under this condition is:
w＝lpm+g+hs(15)
C. when the magnetic flux path is larger than the leakage flux path, leakage flux occurs, and the leakage flux model is as shown in fig. 7. Center point thereof
Is the center point of the spherical surface on the magnetic pole, and the point D is the exit point of the magnetic force line
Point O is the center of the sphere, point G is a point on the edge of the permanent magnet, line OE and line OD form a plane, point G and the leakage path are both on the plane, angle ∠ EOG is:
wherein r isrIs the magnetic pole radius.
Obtaining the following formula according to the cosine of the three-face angle:
and has the following components:
∠DOG＝∠EOG-∠EOD (18)
w＝π|DG|+2lpm(20)
the leakage flux path length can be obtained by equations (16) to (20). Comparing the magnetic flux path length with the permanent magnet leakage path length, if the air gap length is larger than the leakage length, the magnetic flux line is injected into the rotor yoke, no cogging torque is generated, and the air gap length becomes the leakage path length.
According to the above, when the three conditions that the magnetic force lines do not penetrate into the bottom end of the iron tooth, and the length of the magnetic force line penetrating into the iron tooth is smaller than the length of the magnetic force line penetrating into the stator yoke and smaller than the magnetic leakage length are satisfied, the magnetic force lines penetrate into the iron tooth from the side edges, and the side edge force and the cogging torque are generated. And then the magnetic flux density B on each magnetic line generating the side force can be obtained according to the formula (2).
(3) Side force model
The module value of the side force F can be obtained in the formula (3) of solving the cogging torque by the side force as follows:
the infinitesimal area △ S of the magnetic line of force exiting the spherical surface of the magnetic pole is:
wherein △ theta is the magnetic pole according to the latitude direction in theta in figure 21Divide equally NiThe preparation method comprises the following steps of (1),
averaging 2 π to N in the longitudinal direction for a magnetic polejAnd (4) portions are obtained.
Side force
Is a vector, which is divided into F according to the stator coordinate systemX、FY、FZThe three components, as shown in figure 6,
and
in the same direction as
In the same direction, the angle between the point B' and the point B is the same in the spherical coordinate,
the unit vector of (a) is:
then FXThe expression is as follows:
FYthe expression is as follows:
FZthe expression is as follows:
(4) moment arm model
As shown in fig. 6, when the magnetic force line is incident on the iron tooth from point C, the moment arm corresponding to the magnetic force line is OC, the three lines of the straight lines OB, OC and OD are coplanar, point a is a point on the straight line OD, ∠ AOB is obtained from the above-mentioned flux tube path selection, ∠ COB is:
∠ AOC is:
∠AOC＝∠AOB-∠COB (28)
the trigonometric cosine formula was applied to ∠ COB and ∠ AOC:
the point C sphere coordinate can be obtained by the formulas (27) - (30)
Wherein R is:
the moment arm is:
(5) torque model
T, R, F are vectors, T is the cross product of R and F, and Tx is:
ty is:
tz is as follows:
(6) cogging torque synthesis
The invention selects a regular hexahedron rotor structure designed by Japanese T.yano et al, as shown in FIG. 8, eight magnetic poles positioned on the vertex of the regular hexahedron are respectively (R) at initial positions under a stator spherical coordinate systemr,54.74°,45°)、(Rr,54.74°,135°)、(Rr,54.74°,225°)、(Rr,54.74°,315°)、 (Rr,125.26°,45°)、(Rr,125.26°,135°)、(Rr,125.26°,225°)、(Rr125.26 DEG, 315 DEG the rotor coordinate system is rotated as a whole in accordance with Euler rotations z-y-z respectively (α, gamma), each magnetic pole rotates synchronously with the coordinate system under the rotor coordinate system, and the spherical coordinates of the magnetic line emergence points on the magnetic poles change with the rotor rotation as:
wherein (x)r,yr,zr) Is the space coordinate of the magnetic line emergence point under the stator coordinate system before rotation, (x)s,ys,zs) The space coordinate of the exit point of the magnetic line of force under the stator coordinate system after rotation.
The stator adopts a hemispherical shell structure, and the stator tooth structure corresponds to the eight-pole rotor structure in the chapter, so that the amplitude of the coefficient of the primary order harmonic of a rotor magnetic field is larger than that of other order harmonics, the number of coils of the stator tooth structure is the least, and the structure is the simplest. In fig. 8, the number of the spherical motor stator teeth is twelve, the spherical motor stator teeth are arranged in two layers, the number of the first layer is four, the latitudes of the spherical motor stator teeth are the same, the longitude direction difference is 90 degrees, and the space sphere coordinates are as follows: (Rs,30.37 °,45 °), (Rs,30.37 °, 135 °), (Rs,30.37 °,225 °), (Rs,30.37 °,315 °); the second layer is eight, the latitudes of the teeth are the same, the longitude direction is different by 45 degrees, and the space sphere coordinates are as follows: (Rs,68.75 degrees, 22.5 degrees), (Rs,68.75 degrees, 67.5 degrees), (Rs,68.75 degrees, 112.5 degrees), (Rs,68.75 degrees, 157.5 degrees, (Rs,68.75 degrees, 202.5 degrees), (Rs,68.75 degrees, 247.5 degrees), (Rs,68.75 degrees, 292.5 degrees), (Rs,68.75 degrees, 337.5 degrees).
The cogging torque generated by all the magnetic lines on a single magnetic pole is according to TX、TY、TZAnd (3) superposing, namely superposing the cogging torque generated by the whole stator hemisphere for a single magnetic pole after superposition, and superposing the cogging torque generated by the eight magnetic poles to obtain the cogging torque borne by the whole stator hemisphere. Because the action of the forces is mutual, the cogging torque of the whole rotor is the same as the cogging torque of the whole stator in amplitude, and the positive and negative are opposite.
In order to verify the correctness of the proposed equivalent magnetic circuit method for calculating the cogging torque, a hemispherical regular octahedral stator regular hexahedral rotor spherical motor is selected for simulation verification. Firstly, the correctness of the equivalent magnetic circuit method for calculating the cogging torque generated by the monopole rotor is verified, wherein the spherical coordinate of the initial position of the permanent magnet is (0 DEG ), and the spherical coordinate is along
Moving 180 deg., the step size is 3 deg., and its X, Y, Z directional component is shown in fig. 9. The result shows that the method has higher accuracy in calculating the unipolar cogging torque, and it can be seen that the cogging torque generated by the magnetic pole at the moment has an X component in the opposite direction of the same value as the Y component, and a Z component is basically zero. In addition, because the stator is of a hemispherical structure, the cogging torque is rapidly reduced to zero when the rotor magnetic poles move to the lower hemisphere.
The invention then verifies the correctness of the analytic expression of the cogging torque of the multipole rotor calculated by the equivalent magnetic circuit method, wherein the motion track of the multipole rotor is shown in figure 10, the rotor firstly rotates 30 degrees around the z axis, then the y axis of the motion track rotates 180 degrees, and the step length is 5 degrees. As shown in fig. 11, the cogging torque of the spherical motor based on the equivalent magnetic circuit method has the same trend as the simulated value, but the error is larger than that of the monopole, and the spherical motor oscillates at some positions. On the one hand, the equivalent magnetic circuit method ignores the problem of magnetic saturation of individual teeth in the case of multipolar operation, and on the other hand, the torque in the single-pole magnetic circuit method is subjected to rotation conversion and superposition, thereby further amplifying the error. But the equivalent magnetic circuit method is simple and convenient to calculate, the calculation time is greatly shortened compared with the finite element method, and the calculation result still has important reference value in the initial stage of research.
Claims (1)
1. A permanent magnet spherical motor tooth socket torque analysis method based on an equivalent magnetic circuit method comprises the following steps:
the first step is as follows: calculating the air gap flux density generated by a single flux tube: according to the structure of the single-pole multi-tooth permanent magnet spherical motor, a permanent magnet spherical motor tooth-groove torque model is established by adopting a discretization numerical analysis method, a stator winding is not considered, a permanent magnet is selected as a unique magnetomotive force source of the permanent magnet spherical motor, magnetic poles are divided discretely according to spherical coordinates, and an included angle theta 1 of the spherical coordinates occupied by the magnetic poles is divided into N according to the longitude theta directioniIn terms of latitude
Direction divides 2 pi into NjCalculating the initial spherical coordinates of each magnetic flux tube
Comprises the following steps:
wherein r isrIs the radius of the magnetic pole, RrIs the outer diameter of the magnetic pole of the rotor,
as the initial spherical coordinates of the flux tube, (x)0,y0,z0) The initial rectangular coordinate of the magnetic flux tube is shown, subscript i, j is the label of the magnetic flux tube, N+The magnetic flux tube represents a positive integer, magnetic force lines on the permanent magnet are set to be uniformly distributed, and the magnetic flux tube is divided into four conditions according to different inflow positions of the magnetic flux tube: the first is that the magnetic flux tube flows into the side of the stator iron teeth, and the magnetic flux tube generates side force at the moment; the second is that the magnetic flux tube flows into the yoke part of the stator, and the magnetic flux tube does not generate side force at the moment; the third is that the magnetic flux tube flows into the bottom of the stator tooth, and the magnetic flux tube does not generate side force; the fourth is that the magnetic flux tube flows into the rotor ball, and the magnetic flux tube does not generate side force at the moment; the path lengths of the flux tubes at different positions are obtained according to the four conditions, and the air gap flux density B is obtained on the basis
Where μ is the air gap permeability, w is the flux path length in the air gap, H is the magnetic field strength, lpmIs the magnetic pole height;
the second step is that: calculating the cogging torque of the flux tube on the stator teeth: because the magnetic flux tube flows into the side edge of the stator iron tooth to generate the side force and the cogging torque, the cogging torque under the condition is calculated, firstly, the side force F generated by a single magnetic flux tube is calculated by a side force method as follows:
△ S is the infinitesimal area of the magnetic pole sphere ejected by the flux tube;
secondly, obtaining a force arm R of the magnetic flux tube generating cogging torque according to the position of the magnetic flux tube flowing into the side edge of the stator tooth:
wherein R issDenotes the stator tooth inner diameter, rsThe radius of a stator cylinder is defined, and t is the projection length of an emergent point and an incident point of the flux tube on the spherical surface at the bottom of the stator tooth;
finally, the cogging torque T generated by the flux tube acting on the stator teeth is obtained as F multiplied by R, and the cogging torque is decomposed along a rectangular coordinate system to obtain torque components which are respectively as follows:
wherein Fx, Fy and Fz are respectively components of side force under a rectangular coordinate system, Rx, Ry and Rz are respectively components of force arm under the rectangular coordinate system, and the incident point-sphere coordinate of the flux tube is
The coordinates of the emergent point and the sphere of the magnetic flux tube are
The stator tooth spherical coordinate is
∠ AOB is the included angle between the emergent ray of the flux tube and the center line of the stator teeth;
the third step: cogging torque produced by a single pole pair stator tooth: calculating the cogging torque generated by the flux tube according to the magnetic flux tube spherical coordinates, and performing vector superposition on the cogging torque to obtain the cogging torque generated by the single-pole stator teeth;
the fourth step: cogging torque produced by all pole pairs stator teeth: and (2) calculating the cogging torque of each magnetic pole on the stator teeth according to the initial position of each magnetic pole in the permanent magnet spherical motor, and then performing vector superposition on the cogging torque to obtain the cogging torque of the whole stator structure:
wherein, TckCogging torque for the kth pole pair stator teeth, TcogAnd (3) the cogging torque generated by the whole rotor is obtained, n is the number of magnetic poles of the rotor, Euler rotation change is used for expressing the rotation change of the rotor of the spherical motor, and the position of the magnetic poles of the rotor is changed along with the Euler rotation change, so that the cogging torque of the rotor of the spherical motor at any position is obtained.
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