CN113992092B - Surface-mounted permanent magnet motor driving system field path coupling analysis method based on analysis method - Google Patents

Surface-mounted permanent magnet motor driving system field path coupling analysis method based on analysis method Download PDF

Info

Publication number
CN113992092B
CN113992092B CN202111288041.1A CN202111288041A CN113992092B CN 113992092 B CN113992092 B CN 113992092B CN 202111288041 A CN202111288041 A CN 202111288041A CN 113992092 B CN113992092 B CN 113992092B
Authority
CN
China
Prior art keywords
vector
motor
coefficient
matrix
representing
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Active
Application number
CN202111288041.1A
Other languages
Chinese (zh)
Other versions
CN113992092A (en
Inventor
史婷娜
郑碧凝
张振
阎彦
夏长亮
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Zhejiang University ZJU
Zhejiang University Advanced Electrical Equipment Innovation Center
Original Assignee
Zhejiang University ZJU
Zhejiang University Advanced Electrical Equipment Innovation Center
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Zhejiang University ZJU, Zhejiang University Advanced Electrical Equipment Innovation Center filed Critical Zhejiang University ZJU
Priority to CN202111288041.1A priority Critical patent/CN113992092B/en
Publication of CN113992092A publication Critical patent/CN113992092A/en
Application granted granted Critical
Publication of CN113992092B publication Critical patent/CN113992092B/en
Active legal-status Critical Current
Anticipated expiration legal-status Critical

Links

Images

Classifications

    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/14—Estimation or adaptation of machine parameters, e.g. flux, current or voltage
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/14—Estimation or adaptation of machine parameters, e.g. flux, current or voltage
    • H02P21/18—Estimation of position or speed
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P21/00—Arrangements or methods for the control of electric machines by vector control, e.g. by control of field orientation
    • H02P21/14—Estimation or adaptation of machine parameters, e.g. flux, current or voltage
    • H02P21/20—Estimation of torque
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P25/00—Arrangements or methods for the control of AC motors characterised by the kind of AC motor or by structural details
    • H02P25/02—Arrangements or methods for the control of AC motors characterised by the kind of AC motor or by structural details characterised by the kind of motor
    • H02P25/022—Synchronous motors
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P27/00—Arrangements or methods for the control of AC motors characterised by the kind of supply voltage
    • H02P27/04—Arrangements or methods for the control of AC motors characterised by the kind of supply voltage using variable-frequency supply voltage, e.g. inverter or converter supply voltage
    • H02P27/06—Arrangements or methods for the control of AC motors characterised by the kind of supply voltage using variable-frequency supply voltage, e.g. inverter or converter supply voltage using dc to ac converters or inverters
    • H02P27/08—Arrangements or methods for the control of AC motors characterised by the kind of supply voltage using variable-frequency supply voltage, e.g. inverter or converter supply voltage using dc to ac converters or inverters with pulse width modulation
    • H02P27/085—Arrangements or methods for the control of AC motors characterised by the kind of supply voltage using variable-frequency supply voltage, e.g. inverter or converter supply voltage using dc to ac converters or inverters with pulse width modulation wherein the PWM mode is adapted on the running conditions of the motor, e.g. the switching frequency
    • H—ELECTRICITY
    • H02—GENERATION; CONVERSION OR DISTRIBUTION OF ELECTRIC POWER
    • H02P—CONTROL OR REGULATION OF ELECTRIC MOTORS, ELECTRIC GENERATORS OR DYNAMO-ELECTRIC CONVERTERS; CONTROLLING TRANSFORMERS, REACTORS OR CHOKE COILS
    • H02P2207/00—Indexing scheme relating to controlling arrangements characterised by the type of motor
    • H02P2207/05—Synchronous machines, e.g. with permanent magnets or DC excitation
    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
    • Y02—TECHNOLOGIES OR APPLICATIONS FOR MITIGATION OR ADAPTATION AGAINST CLIMATE CHANGE
    • Y02T—CLIMATE CHANGE MITIGATION TECHNOLOGIES RELATED TO TRANSPORTATION
    • Y02T10/00—Road transport of goods or passengers
    • Y02T10/60—Other road transportation technologies with climate change mitigation effect
    • Y02T10/64—Electric machine technologies in electromobility

Landscapes

  • Engineering & Computer Science (AREA)
  • Power Engineering (AREA)
  • Tests Of Circuit Breakers, Generators, And Electric Motors (AREA)
  • Control Of Ac Motors In General (AREA)

Abstract

The invention discloses a field path coupling analysis method of a surface-mounted permanent magnet motor driving system based on an analysis method. According to a voltage balance equation of the motor, a coupling relation of a motor line voltage vector, a phase current vector and a stator winding flux linkage vector is established; establishing a coupling relation among a phase current vector of the motor, a stator winding flux linkage vector and motor structural parameters by a magnetic field analysis calculation method; establishing an analytic relational expression of motor line voltage vectors, phase current vectors and motor structural parameters; constructing a lumped solving matrix; and solving the lumped solution matrix, and simultaneously obtaining all coefficient vectors to be determined in the general solution expression of the phase current vector and the vector magnetic potential of the current time step, so as to obtain and store the stator winding flux linkage vector of the current time step, and further analyzing the output torque and the torque fluctuation rate of the motor. The method can calculate the phase current and the vector magnetic position simultaneously, has good calculation instantaneity, and can effectively improve the calculation efficiency on the premise of ensuring the calculation accuracy.

Description

Surface-mounted permanent magnet motor driving system field path coupling analysis method based on analysis method
Technical Field
The invention relates to a field coupling analysis method of a permanent magnet motor driving system, in particular to a field coupling analysis method of a double-layer concentrated winding surface-mounted permanent magnet motor driving system based on an analysis method.
Background
The permanent magnet motor has the advantages of high power density, high operation efficiency and the like, and is widely applied to the fields of industrial robots, numerical control machine tools, electric automobiles and the like. The pulse width modulation technology is an important technical means for realizing stable operation of the permanent magnet motor, but the chopper of the inverter switching tube can cause sideband harmonic components near carrier frequency or multiple frequency thereof to exist in motor phase current; on the other hand, the space harmonics of the flux linkage introduce low frequency harmonic components for the phase currents, which are affected by the motor body structure. These phase current time harmonics all affect important performance parameters such as the output torque of the motor. By using the field coupling method, unified modeling of the inverter and the permanent magnet motor can be realized, and the performance of the permanent magnet motor can be accurately analyzed.
The existing field coupling method can be divided into two types of indirect coupling and direct coupling. The magnetic field calculation and the circuit calculation of the indirect coupling method are carried out in two steps, so that a time step delay is inevitably existed between calculation results of the two steps, and the system is unstable when serious; in addition, when used in the initial optimization design stage of the motor requiring frequent structural parameter change, the indirect coupling method requires repeating the establishment process of the circuit parameter table for a plurality of times by using the magnetic field calculation model, and has complex steps and low efficiency. The direct coupling method can calculate the magnetic field quantity and the electric quantity simultaneously, the real-time performance of the calculation result is good, and the accuracy is high, but the existing direct coupling method is based on a finite element-based magnetic field calculation model (called a field path coupling time-step finite element model), and the calculation efficiency cannot be simultaneously ensured while the calculation accuracy is ensured.
Disclosure of Invention
In order to realize the rapid and accurate analysis of the performance parameters of the double-layer concentrated winding surface-mounted permanent magnet motor, the invention provides the idea of simultaneously analyzing and solving the magnetic field quantity and the electric quantity, and an analysis relation of motor line voltage, phase current and motor structural parameters is established in three steps. And the method realizes simultaneous analytic calculation of vector magnetic potential and phase current by constructing a field path coupling analytic model lumped solving matrix with motor line voltage as an excitation source. And further, under the condition of considering the driving parameters and the influence factors of the motor structural parameters, the accurate analysis of the motor performance is realized.
The invention adopts the following technical scheme:
the invention comprises the following steps:
1) For a permanent magnet motor with three-phase windings in a star connection mode, establishing a coupling relation of a motor line voltage vector, a phase current vector and a stator winding flux linkage vector according to a voltage balance equation of the permanent magnet motor;
2) Establishing a coupling relation among a phase current vector of the permanent magnet motor, a stator winding flux linkage vector and motor structural parameters by a magnetic field analysis calculation method;
3) Establishing an analytic relational expression of the motor line voltage vector, the phase current vector and the motor structural parameter according to the coupling relation of the motor line voltage vector, the phase current vector and the stator winding flux linkage vector and the coupling relation of the phase current vector, the stator winding flux linkage vector and the motor structural parameter of the permanent magnet motor;
4) Establishing constraint equations satisfied by first-ninth undetermined coefficients and phase currents in the magnetic flux-position solution expressions of all sub-field vectors by utilizing interface boundary conditions of continuous magnetic density radial components and continuous magnetic field intensity tangential components between all adjacent sub-fields, and constructing a lumped solution matrix by combining analytic relational expressions of motor line voltage vectors, phase current vectors and motor structural parameters in the step 3);
5) In each time step, according to the motor line voltage vector of the current time step, the phase current vector of the previous time step, the stator winding flux linkage vector and the structural parameters of the motor, solving a lumped solving matrix, simultaneously obtaining the phase current vector of the current time step and the first-ninth undetermined coefficient vector in the general solution expression of the flux position of each sub-field vector, according to the coupling relation of the phase current vector of the motor, the stator winding flux linkage vector and the structural parameters of the motor in the step 2), calculating to obtain the stator winding flux linkage vector of the current time step, and storing the phase current vector of the current time step and the stator winding flux linkage vector of the current time step for calculating the next time step;
6) And calculating the radial component and tangential component of the magnetic density of the air gap subdomain according to the general solution expression of the vector magnetic bits of the air gap subdomain in each subdomain, and further analyzing the output torque and the torque fluctuation rate of the motor.
The coupling relation among the motor line voltage vector, the phase current vector and the stator winding flux linkage vector in the step 1) satisfies the following formula:
Figure GDA0004192900430000021
wherein t is s Is the time step; u (u) L ＝[u ab u bc ] T Line voltage vector representing current time step, u ab And u bc Line voltages between a phase and b phase and between b phase and c phase respectively; i.e ab And psi respectively represents a phase current vector of the current time step and a flux linkage vector of the stator winding, thereby meeting i ab ＝[i a i b ] T ，i a ，i b Respectively representing a phase current and a phase current of b phase current; ψ= [ ψ ] a ψ b ψ c ] T ，ψ a ，ψ b ，ψ c Respectively a, b and c phase winding flux linkage, wherein T represents matrix transposition operation; i.e ab # Psi (phi) # Phase current vectors and stator winding flux linkage vectors respectively representing the previous time step; r is phase resistance, L 0 Is leakage inductance; c (C) + A first coefficient matrix representing a voltage balance equation, satisfying
Figure GDA0004192900430000022
C - A second coefficient matrix representing a voltage balance equation, satisfying +.>
Figure GDA0004192900430000023
The step 2) specifically comprises the following steps:
2.1 Simplifying a magnetic field solving model of the motor into a two-dimensional model, and dividing a magnetic field solving area of the two-dimensional model into areas to obtain all the subareas, wherein each subarea comprises a permanent magnet subarea, an air gap subarea, a slot subarea and a slot subarea;
2.2 Solving partial differential equations satisfied by the vector magnetic bits of all the subfields by using a separation variable method to obtain a general solution expression of the vector magnetic bit equation of all the subfields, and further obtaining flux linkage of each layer of winding turn chain in the slot subfields after integrating the vector magnetic bits of the slot subfields;
2.3 Presetting an incidence matrix of stator windings in Q slot subdomains, obtaining an expression of a stator winding flux linkage vector according to flux linkage of each layer of winding turn linkage in each slot and a corresponding incidence matrix, and using the expression as a coupling relation of a phase current vector of a motor, the stator winding flux linkage vector and motor structural parameters, wherein the coupling relation satisfies the following formula:
Figure GDA0004192900430000031
wherein A is 2 For the third coefficient vector to be determined, B 2 For the fourth coefficient vector to be determined, C 2 For the fifth coefficient vector to be determined, D 2 For the sixth coefficient vector to be determined, D 3t Is a ninth undetermined coefficient vector; ρ c1 Representing the first flux linkage coefficient to satisfy ρ c1 ＝lN c b sa /S h ，ρ c2 Representing the second flux linkage coefficient, satisfying ρ c2 ＝μ 0 N c /N a /S h ，ρ c3 Representing the third flux linkage coefficient, satisfying ρ c3 ＝(R sb 2 -R t 2 )/4，ρ c4 Represents the fourth flux linkage coefficient, satisfies
Figure GDA0004192900430000032
Q is the number of stator slots, b oa For wide angle of notch b sa For the stator slot width angle, N c For the number of turns of each coil S h For the area occupied by each layer of winding in the slot, N a The number of parallel branches is l, the effective length of the motor is mu 0 Is air permeability, R r 、R m 、R s 、R t 、R sb Respectively representing the outer radius of the rotor, the outer radius of the permanent magnet, the inner radius of the stator, the arc radius of the groove top and the arc radius of the groove bottom; ρ x2s Representing a sixth flux linkage coefficient matrix,>
Figure GDA0004192900430000033
a seventh flux linkage coefficient matrix representing a sub-field of the q-th slot, satisfying +.>
Figure GDA0004192900430000034
diag () represents a diagonal matrix; ρ x3s Represents an eighth flux linkage coefficient matrix,>
Figure GDA0004192900430000035
a ninth flux linkage coefficient matrix representing a sub-field of the q-th slot satisfying +.>
Figure GDA0004192900430000036
ρ x5s Representing an eleventh flux linkage coefficient matrix,>
Figure GDA0004192900430000037
a twelfth flux linkage coefficient matrix representing a q-th slot subfield satisfying
Figure GDA0004192900430000038
ρ x6 Representing a thirteenth flux linkage coefficient matrix satisfying ρ x6 ＝[(L K σ 10 ) T ,(L K σ 20 ) T ,...,(L K σ Q0 ) T ] T Q×K ，ρ x7 Represents a fourteenth flux linkage coefficient matrix satisfying ρ x7 ＝[(L K τ 10 ) T ,(L K τ 20 ) T ,...,(L K τ Q0 ) T ] T Q×K K is the harmonic frequency of the magnetic field distribution in the permanent magnet sub-field and the air gap sub-field, the maximum value of which is represented by K, L K Representing dimension k=k max Line vector L of (1) K ＝(1) 1×K ；G 2 Is a second motor structural parameter matrix, G 2k Is the structural parameter coefficient of the kth second motor, meets G 2 ＝diag(G 21 ,G 22 ,…,G 2K )，G 2k ＝(R m /R s ) k ；I Q An identity matrix with dimension Q; sigma (sigma) q0 Fifth motor structural parameter matrix representing a q-th slot subfield，σ q0 (k) The kth fifth motor structural parameter coefficient representing the qth slot subdomain satisfies sigma q0 ＝diag(σ q0 (1),σ q0 (2),...,σ q0 (K)) K×K ，σ q0 (k)＝2sin(kb oa /2)cos(kα q )/(kb oa )；α q The groove center position angle of the q-th groove sub-field; τ q0 Sixth motor structural parameter matrix, τ, representing a q-th slot sub-field q0 (k) The kth and sixth motor structural parameter coefficients representing the qth slot subdomain satisfy τ q0 ＝diag(τ q0 (1),τ q0 (2),...,τ q0 (K)) K×K ,τ q0 (k)＝2sin(kb oa /2)sin(kα q )/(kb oa )；
Figure GDA0004192900430000041
Figure GDA0004192900430000042
Figure GDA0004192900430000043
Wherein ρ is x1 Representing a fifth flux linkage coefficient matrix satisfying ρ x1 ＝diag(sin(π/2),sin(π),...,sin(Nπ/2))；ρ x4 Represents a tenth flux linkage coefficient matrix satisfying ρ x4 ＝diag(cos(π),cos(2π),...,cos(Nπ))；ζ 0 Representing a seventh motor structural parameter matrix, ζ 0 (n) represents the nth seventh motor structural parameter coefficient satisfying ζ 0 ＝diag(ζ 0 (1),ζ 0 (2),...,ζ 0 (N)) N×N ,ζ 0 (n)＝2cos(nπ2)sin(E n b oa /2)/(nπγ a )，γ a Is the ratio of the width of the groove opening to the width of the groove, and satisfies gamma a ＝b oa /b sa ；E n A harmonic matrix distributed for the slot subdomain magnetic field, satisfies E n ＝diag(E 1 ,E 2 ,...,E N )，E n The harmonic frequency of the magnetic field distribution of the groove subdomain is expressed, and E is satisfied n ＝nπ/b sa N represents the harmonic frequency coefficient of the slot subfield magnetic field distribution, n=1, 2,3 …, the maximum of which is represented by N; l (L) N Representing the dimension n=n max Line vector L of (1) N ＝(1) 1×N ；I N For dimension n=n max Is a matrix of units of (a); g 3 Is a third motor structure parameter matrix, meets G 3 ＝diag(G 31 ,G 32 ,…,G 3N )，G 3n Representing the structural parameter coefficient of the nth third motor, G 3n ＝(R t /R sb ) En ；N＝diag(1,2,…,N)。
The conditions for simplifying the electromagnetic field of the motor in the step 2.1) are specifically as follows:
1. the magnetic conductivity of the stator and rotor iron core is infinity; 2. the demagnetization curve of the permanent magnet material is linear; 3. ignoring the conductivity effect and eddy current effect of the ferromagnetic material; 4. the vector magnetic bits of each subdomain only contain axial components; 5. the structure of the groove and the notch is simplified into a fan-shaped structure.
The correlation matrix C in the step 2.3) x The following formula is satisfied:
Figure GDA0004192900430000051
wherein s is aq,x ，s bq,x Sum s cq,x Respectively represent the reference directions of a, b and c phase currents in the x-th layer winding in the Q-th slot, q=1, 2, …, Q, x=i, II and s aq,x ，s bq,x Sum s cq,x The value of 1 indicates that current is flowing, -1 indicates that current is flowing, and 0 indicates that there is no phase current.
The analytic relation of the motor line voltage vector, the phase current vector and the motor structural parameters in the step 3) meets the following formula:
K 103 A 2 +K 104 B 2 +K 105 C 2 +K 106 D 2 +K 109 D 3t +K 1010 i ab ＝Y 10
wherein,,
K 103 ＝ρ c1 ρ c3 C + (C I +C II )ρ x6 /N a ；
K 104 ＝ρ c1 ρ c3 C + (C I +C II )ρ x6 G 2 /N a ；
K 105 ＝ρ c1 ρ c3 C + (C I +C II )ρ x7 /N a ；
K 106 ＝ρ c1 ρ c3 C + (C I +C II )ρ x7 G 2 /N a ；
K 109 ＝C + [C I (ρ x2s -ρ x3s )-C II (ρ x2s +ρ x3s )]/N a ；
Figure GDA0004192900430000052
Figure GDA0004192900430000053
K 103 a first coefficient matrix, K, representing a variant of the voltage balance equation 104 A second coefficient matrix, K, representing a variant of the voltage balance equation 105 A third coefficient matrix, K, representing a variant of the voltage balance equation 106 A fourth coefficient matrix, K, representing a variant of the voltage balance equation 109 A fifth coefficient matrix, K, representing a variant of the voltage balance equation 1010 A sixth coefficient matrix representing a variant of the voltage balance equation, Y 10 A seventh coefficient matrix representing a variation of the voltage balance equation.
The step 4) is specifically as follows:
according to the boundary conditions of the interface of the notch subdomain and the slot subdomain, obtaining:
K 87 C 4t +K 88 D 4t +K 89 D 3t +K 810 i ab ＝0
K 97 C 4t +K 98 D 4t +K 99 D 3t +K 910 i ab ＝0
and obtaining according to boundary conditions of interfaces of the notch subdomain and the air gap subdomain:
K 53 A 2 +K 54 B 2 +K 57 C 4t +K 58 D 4t +K 510 i ab ＝0
K 65 C 2 +K 66 D 2 +K 67 C 4t +K 68 D 4t +K 610 i ab ＝0
K 73 A 2 +K 74 B 2 +K 75 C 2 +K 76 D 2 +K 77 C 4t +K 78 D 4t ＝0
according to the boundary conditions of the interfaces of the permanent magnet subdomains and the air gap subdomains, obtaining:
K 11 A 1 +K 13 A 2 +K 14 B 2 ＝Y 1
K 22 C 1 +K 25 C 2 +K 26 D 2 ＝Y 2
K 31 A 1 +K 33 A 2 +K 34 B 2 ＝Y 3
K 42 C 1 +K 45 C 2 +K 46 D 2 ＝Y 4
wherein A is 1 For the first coefficient vector to be determined, C 1 For the second coefficient vector to be determined, C 4t For the seventh coefficient vector to be determined, D 4t Is an eighth undetermined coefficient vector; k (K) 11 -K 910 Y is as follows 1 -Y 4 First-fourth coefficient matrixes of the lumped solving matrix of the field path coupling analytic model are respectively; namely K 11 First coefficient matrix, K, of lumped solution matrix for field-path coupling analytical model 13 -K 14 Respectively isSecond-third coefficient matrix of lumped solution matrix of field path coupling analytical model, K 22 Fourth coefficient matrix, K, of lumped solution matrix for field-path coupling analytical model 25 -K 26 Fifth-sixth coefficient matrix, K, of lumped solution matrix of field-path coupling analytical model respectively 31 Seventh coefficient matrix, K, of lumped solution matrix for field-path coupling analytical model 33 -K 34 Eighth-ninth coefficient matrix, K, of lumped solution matrix of field path coupling analytical model respectively 42 Tenth coefficient matrix, K, of lumped solution matrix for field-path coupling analytical model 45 -K 46 Eleventh-twelfth coefficient matrix, K, of lumped solution matrix of field-path coupling analytical model respectively 53 -K 54 Thirteenth-fourteenth coefficient matrix, K, of lumped solution matrix of field-path coupling analytical model, respectively 57 -K 58 Fifteenth-sixteenth coefficient matrix, K, of lumped solution matrix of field-path coupling analytical model, respectively 510 Seventeenth coefficient matrix, K, of lumped solution matrix for field-path coupling analytical model 65 -K 68 Eighteenth-twentieth coefficient matrix, K, of lumped solution matrix of field-path coupling analytical model, respectively 610 Twenty-second coefficient matrix, K, of lumped solution matrix for field-path coupling analytical model 73 -K 78 Twenty-third-twenty-eighth coefficient matrix, K, of lumped solution matrix of field path coupling analytical model respectively 87 -K 810 Second nineteenth-thirty second coefficient matrix, K, of lumped solution matrix of field-path coupling analytical model respectively 97 -K 910 Thirteenth-thirty-sixth coefficient matrix of lumped solution matrix of field path coupling analytical model respectively, Y 1 -Y 4 A seventeenth-forty coefficient matrix of the lumped solution matrix of the field path coupling analytic model respectively; k (K) 11 -K 910 Y is as follows 1 -Y 4 Is a constant matrix related to motor structural parameters and permanent magnet magnetization;
Figure GDA0004192900430000061
Figure GDA0004192900430000062
Figure GDA0004192900430000063
Figure GDA0004192900430000064
wherein K is 810_1 For the first matrix of boundary condition coefficients,
Figure GDA0004192900430000065
for the second boundary condition coefficient matrix of the q-th slot sub-field, satisfy +.>
Figure GDA0004192900430000066
K 910_1 For the third boundary condition coefficient matrix, ++>
Figure GDA0004192900430000067
Fourth boundary condition coefficient matrix for the q-th slot subdomain, satisfy +.>
Figure GDA0004192900430000068
K 910_2 For the fifth boundary condition coefficient matrix, +.>
Figure GDA0004192900430000069
A sixth boundary condition coefficient matrix for the q-th slot subdomain, satisfies
Figure GDA0004192900430000071
η 0 Represents an eighth motor structural parameter matrix, satisfies the following conditions
Figure GDA0004192900430000072
η q0 Represents a ninth motor structural parameter matrix, eta q0 (k) Represents the kth ninthThe parameter coefficient of the motor structure meets eta q0 ＝(η q0 (k)) 1×K The method comprises the steps of carrying out a first treatment on the surface of the Xi 0 represents tenth motor structural parameter matrix satisfying +.>
Figure GDA0004192900430000073
ξ q0 Representing an eleventh motor structural parameter matrix, xi q0 (k) Represents the structural parameter coefficient of the kth and eleventh motor, satisfies the requirement of xi q0 ＝(ξ q0 (k)) 1×K ；
Figure GDA0004192900430000074
Figure GDA0004192900430000075
Figure GDA0004192900430000076
Wherein, gamma 0 Representing a twelfth motor structural parameter matrix, gamma 0 (n) represents the nth twelfth motor structural parameter coefficient satisfying γ 0 ＝(γ 0 (n)) 1×N The method comprises the steps of carrying out a first treatment on the surface of the ζ is a thirteenth motor structural parameter matrix, satisfying ζ=γ/γ a Gamma represents a fourteenth motor structural parameter matrix, gamma (m, n) represents a fourteenth motor structural parameter coefficient when the harmonic order coefficient of the slot sub-field distribution is m and the harmonic order coefficient of the slot sub-field distribution is n, satisfying gamma= (gamma (m, n)) M×N ；
γ 0 (n)＝4cos(nπ/2)sin(E n b oa /2)/nπ；
Figure GDA0004192900430000077
Figure GDA0004192900430000078
Figure GDA0004192900430000079
Wherein alpha is q The central position angle of each groove; f (F) m The harmonic frequency of the notch subdomain magnetic field distribution is satisfied with F m ＝mπ/b oa M represents the harmonic frequency coefficient of the notch subfield magnetic field distribution, satisfying m=1, 2,3 …, the maximum value of which is represented by M;
the lumped solution matrix of the field coupling analytical model is obtained by the method:
Figure GDA0004192900430000081
the beneficial effects of the invention are as follows:
(1) The invention provides the concept of simultaneous analytic solution of magnetic field quantity and electric quantity, and the simultaneous analytic solution of vector magnetic potential and phase current can be realized by constructing a field path coupling analytic model to lumped solve a matrix. Therefore, errors generated by asynchronous calculation of the magnetic field quantity and the electric quantity in the indirect coupling method can be avoided, and the problem that the calculation time of the existing direct coupling method is long is fundamentally solved.
(2) When the lumped solution matrix of the field path coupling analytical model is constructed, the analytical relation of motor line voltage, phase current and motor structural parameters is established. Compared with a field coupling time-step finite element model based on numerical calculation, the field coupling analysis model can clearly and intuitively describe the coupling relation among motor line voltage, phase current and motor structural parameters, and can provide theoretical basis for the parameterization analysis process of the permanent magnet motor driving system.
(3) When the model is provided for calculating the motor performance parameters, the influence of the motor structural parameters can be considered, and the influence factors of the control parameters can be considered according to the change of different control parameters (such as control frequency and modulation ratio) on the motor line voltage vector. By further combining a motor mechanical equation and different control strategies, the model can be used for analyzing the influence rules of control parameters and motor structural parameters on the aspects of motor maneuver, steady-state performance, vibration noise, loss, temperature rise and the like, and lays a foundation for realizing the overall optimization design of the permanent magnet motor driving system.
Drawings
FIG. 1 is a schematic diagram of a surface mount permanent magnet motor drive system;
FIG. 2 is a schematic diagram of slot type parameters and winding current;
FIG. 3 is a block diagram of a field coupling resolution model;
FIG. 4 is a phase a current waveform of a permanent magnet motor;
FIG. 5 is a waveform of a phase flux linkage of a permanent magnet motor;
FIG. 6a is a waveform of the radial component of the air gap flux density of a permanent magnet motor;
FIG. 6b is a waveform of the tangential component of the air gap flux density of the permanent magnet motor;
fig. 7 is a waveform of output torque of the permanent magnet motor.
Detailed Description
The invention is further described with reference to the drawings and specific examples.
Application example of a complete implementation of the inventive content according to the present invention:
in order to establish the field coupling analysis model, the basic parameters of the required double-layer concentrated winding surface-mounted permanent magnet motor are shown in table 1.
Table 1 table-mounted permanent magnet machine basic parameters
Figure GDA0004192900430000091
Embodiments are described below in conjunction with fig. 1-7, and specific calculation formulas, and are described in detail below:
as shown in fig. 3, the present invention includes the steps of:
1) For a permanent magnet motor with three-phase windings in a star connection mode, establishing a coupling relation of a motor line voltage vector, a phase current vector and a stator winding flux linkage vector according to a voltage balance equation of the permanent magnet motor;
in the step 1), the coupling relation among the motor line voltage vector, the phase current vector and the stator winding flux linkage vector satisfies the following formula:
Figure GDA0004192900430000092
wherein t is s Is the time step; u (u) L ＝[u ab u bc ] T Line voltage vector representing current time step, u ab And u bc Line voltages between a phase and b phase and between b phase and c phase respectively; i.e ab And psi respectively represents a phase current vector of the current time step and a flux linkage vector of the stator winding, thereby meeting i ab ＝[i a i b ] T ，i a ，i b Respectively representing a phase current and a phase current of b phase current; ψ= [ ψ ] a ψ b ψ c ] T ，ψ a ，ψ b ，ψ c Respectively a, b and c phase winding flux linkage, wherein T represents matrix transposition operation; i.e ab # Psi (phi) # Phase current vectors and stator winding flux linkage vectors respectively representing the previous time step; r is phase resistance, L 0 Is leakage inductance; c (C) + A first coefficient matrix representing a voltage balance equation, satisfying
Figure GDA0004192900430000101
C - A second coefficient matrix representing a voltage balance equation, satisfying +.>
Figure GDA0004192900430000102
Specifically, the voltage balance equation for a three-phase permanent magnet motor can be expressed as:
Figure GDA0004192900430000103
wherein u is NO Is neutral point voltage; u (u) aO ，u bO ，u cO Terminal voltages of a, b and c phases respectively; i.e a ，i b ，i c Respectively the two groups of the two groups are a,b, c phase current; psi phi type a ，ψ b ，ψ c And a, b and c phase winding flux linkages are respectively adopted.
When the three-phase winding of the permanent magnet motor adopts a star connection mode and the neutral point is not led out, according to kirchhoff current law, the three-phase current meets i a +i b +i c =0. Discretizing the voltage balance equation by utilizing the backward differential pair, and obtaining a matrix form of a motor line voltage equation, wherein the matrix form is as follows:
Figure GDA0004192900430000104
2) Establishing a coupling relation among a phase current vector of the motor, a stator winding flux linkage vector and motor structural parameters by a magnetic field analysis calculation method;
the step 2) is specifically as follows:
2.1 Simplifying a magnetic field solving model of the motor into a two-dimensional model, and obtaining all the subfields after carrying out regional division on the magnetic field solving region of the two-dimensional model according to the structural shape of the motor and the medium properties of different magnetic field regions, wherein each subfield comprises a permanent magnet subfield, an air gap subfield, a slot subfield and a notch subfield; as shown in fig. 1, region 1: the permanent magnet subdomain is an area where the permanent magnet is located, and does not comprise an area where the rotor iron core and the rotating shaft are located; region 2: an air gap subdomain which is an annular gap between the inner ring of the stator and the permanent magnet; region 3q: a q-th slot subdomain, a plurality of slots for placing windings are formed in the stator, and the q-th slot subdomain is arranged in the area where the q-th slot is positioned; and the region 4q and the q-th notch subdomains, the slots for placing windings are provided with notches communicated with the inner ring of the stator, and the region where the q-th notch is positioned is used as the q-th notch subdomain.
The conditions for simplifying the electromagnetic field of the motor in step 2.1) are specifically:
1. the magnetic conductivity of the stator and rotor iron core is infinity; 2. the demagnetization curve of the permanent magnet material is linear; 3. ignoring the conductivity effect and eddy current effect of the ferromagnetic material; 4. the vector magnetic bits of each subdomain only contain axial components; 5. the structure of the groove and the notch is simplified into a fan-shaped structure.
2.2 Solving partial differential equations satisfied by the vector magnetic bits of all the subfields by using a separation variable method to obtain a general solution expression of the vector magnetic bit equation of all the subfields, and further obtaining flux linkage of each layer of winding turn chain in the slot subfields after integrating the vector magnetic bits of the slot subfields;
vector magnetic bit A of permanent magnet subdomain z1 The following partial differential equation is satisfied:
Figure GDA0004192900430000111
wherein mu 0 Is air permeability; r and alpha are the radial position and tangential position of any point in space under the polar coordinate system respectively; m is M r 、M α The radial component and the tangential component of the residual magnetic density after the permanent magnet is magnetized are respectively represented, and fourier decomposition is carried out on the radial component and the tangential component, so that the obtained expression is:
Figure GDA0004192900430000112
wherein k is the harmonic frequency of the magnetic field distribution in the permanent magnet and the air gap; m is M rck 、M rsk Respectively M r Cosine coefficient and sine coefficient after Fourier decomposition; m is M αck 、M αsk Respectively M α Cosine coefficients and sine coefficients after fourier decomposition.
Vector magnetic position A of air gap and notch subdomain zu The following partial differential equation is satisfied:
Figure GDA0004192900430000113
where v is 2 or 4Q, q=1, 2, …, Q.
Vector magnetic bit A of slot subdomain z3q The following partial differential equation is satisfied:
Figure GDA0004192900430000114
wherein J in q slot q For the double layer concentrated winding shown in FIG. 2, the current density J is q After fourier decomposition, it can be expressed as:
Figure GDA0004192900430000115
wherein alpha is q A central position angle for each slot; alpha q -b sa /2≤α≤α q +b sa /2；E n The harmonic frequency of the magnetic field distribution of the groove subdomain is expressed, and E is satisfied n ＝nπ/b sa ，n＝1,2,3…,N，b sa Is the stator slot width angle; n (N) c Turns for each coil; s is S h The occupied area of each layer of winding in the slot is that; n (N) a The number of parallel branches is the number of parallel branches; i.e q,I And i q,II The current flowing through the two layers of windings in the q-th slot respectively meets the following conditions:
Figure GDA0004192900430000116
Figure GDA0004192900430000117
wherein C is I And C II The corresponding incidence matrixes of the windings of the layer I and the layer II in each slot are respectively adopted, and Q is the number of stator slots. For the 10p12s double-layer concentrated winding permanent magnet motor shown in Table 1, C I And C II Respectively is
Figure GDA0004192900430000121
Solving vector magnetic bit A of permanent magnet subdomain, air gap subdomain, slot subdomain and notch subdomain by using separation variable method z1 、A z2 、A z3q 、A z4q The partial differential equation is satisfied, and the general solution expressions of the vector magnetic potential equation in each subdomain are respectively:
Figure GDA0004192900430000122
Figure GDA0004192900430000123
/>
Figure GDA0004192900430000124
Figure GDA0004192900430000125
wherein,,
Figure GDA0004192900430000126
Figure GDA0004192900430000127
a z3q_3 ＝cos[E n (α+b sa /2-α q )]；
a z4q_1 ＝cos[F m (α+b oa /2-α q )]；
ζ 0 (n)＝2cos(nπ/2)sin(E n b oa /2)/(nπγ a )；
σ q0 (k)＝2sin(kb oa /2)cos(kα q )/(kb oa )；
τ q0 (k)＝2sin(kb oa /2)sin(kα q )/(kb oa )；
R r 、R m 、R s 、R t 、R sb respectively representing the outer radius of the rotor, the outer radius of the permanent magnet, the inner radius of the stator, the arc radius of the groove top and the arc radius of the groove bottom; b oa Is a notch wide angle; f (F) m For dividing notch sub-fieldHarmonic frequency of cloth satisfying F m ＝mπ/b oa (m=1, 2,3, …), m representing the harmonic order coefficient of the notch subfield magnetic field distribution; g 1 For the first motor structural parameter matrix, G 1k Is the structural parameter coefficient of the kth first motor, meets G 1 ＝diag(G 11 ,G 12 ,…,G 1K )，G 1k ＝(R r /R m ) k ；a z3q_1 For the first vector magnetic bit flux solution coefficient, a z3q_2 For the second vector magnetic bit flux solution coefficient, a z3q_3 For the third vector magnetic bit to solve for coefficient, a z4q_1 A fourth vector magnetic bit-passing solution coefficient; a is that 1k 、C 1k 、A 2k 、B 2k 、C 2k 、D 2 k 、C 4qm 、D 4qm 、D 3qn The first through ninth coefficients are defined respectively.
2.3 Presetting an incidence matrix of stator windings in Q slot subdomains, obtaining an expression of a stator winding flux linkage vector according to flux linkage of each layer of stator winding turn linkage in each slot and a corresponding incidence matrix, and using the expression as a coupling relation of a phase current vector of a motor, the stator winding flux linkage vector and motor structural parameters, wherein the coupling relation satisfies the following formula:
Figure GDA0004192900430000131
wherein A is 2 For the third coefficient vector to be determined, satisfy A 2 ＝[A 21 ,A 22 ,…,A 2K ] T Is determined by a third coefficient A 2k Component column vectors, B 2 For the fourth coefficient vector to be determined, satisfy B 2 ＝[B 21 ,B 22 ,…,B 2K ] T Is determined by a fourth coefficient B 2k Component column vectors, C 2 For the fifth undetermined coefficient vector, satisfy C 2 ＝[C 21 ,C 22 ,…,C 2K ] T Is composed of the fifth undetermined coefficient C 2k Component column vectors, D 2 For the sixth coefficient vector to be determined, satisfy D 2 ＝[D 21 ,D 22 ,…,D 2K ] T Is formed by the sixth stepCoefficient of undetermining D 2k Component column vectors, D 3t For the ninth coefficient vector to be determined, D 3t To be determined by the ninth undetermined coefficient D 3qn Component column vectors, D 3t ＝[D 31 ,D 32 ,…,D 3Q ] T ，D 3q ＝[D 3q1 ,D 3q2 ,…,D 3qN ] T 。；ρ c1 Representing the first flux linkage coefficient to satisfy ρ c1 ＝lN c b sa /S h ，ρ c2 Representing the second flux linkage coefficient, satisfying ρ c2 ＝μ 0 N c /N a /S h ，ρ c3 Representing the third flux linkage coefficient, satisfying ρ c3 ＝(R sb 2 -R t 2 )/4，ρ c4 Represents the fourth flux linkage coefficient, satisfies
Figure GDA0004192900430000132
The motor structural parameters comprise the number of stator slots, the wide angle of the slot opening, the wide angle of the stator slots, the number of turns of each coil, the occupied area of each layer of windings in the slots, the number of parallel branches, the effective length of the motor, the outer radius of a rotor, the outer radius of a permanent magnet, the inner radius of the stator, the radius of a slot top arc and the radius of a slot bottom arc, Q is the number of stator slots, the number of stator slots is the same as the number of slot subdomains, b oa For wide angle of notch b sa For the stator slot width angle, N c For the number of turns of each coil S h For the area occupied by each layer of winding in the slot, N a The number of parallel branches is l, the effective length of the motor is mu 0 Is air permeability, R r 、R m 、R s 、R t 、R sb Respectively representing the outer radius of the rotor, the outer radius of the permanent magnet, the inner radius of the stator, the arc radius of the groove top and the arc radius of the groove bottom; ρ x2s Representing a sixth flux linkage coefficient matrix,>
Figure GDA0004192900430000133
a seventh flux linkage coefficient matrix representing a sub-field of the q-th slot, satisfying +.>
Figure GDA0004192900430000134
diag () represents a diagonal matrix; ρ x3s Represents an eighth flux linkage coefficient matrix,>
Figure GDA0004192900430000141
a ninth flux linkage coefficient matrix representing a sub-field of the q-th slot satisfying +.>
Figure GDA0004192900430000142
ρ x5s An eleventh matrix of flux linkage coefficients is represented,
Figure GDA0004192900430000143
a twelfth flux linkage coefficient matrix representing a q-th slot subfield satisfying +.>
Figure GDA0004192900430000144
ρ x6 Representing a thirteenth flux linkage coefficient matrix satisfying ρ x6 ＝[(L K σ 10 ) T ,(L K σ 20 ) T ,...,(L K σ Q0 ) T ] T Q×K ，ρ x7 Represents a fourteenth flux linkage coefficient matrix satisfying ρ x7 ＝[(L K τ 10 ) T ,(L K τ 20 ) T ,...,(L K τ Q0 ) T ] T Q×K K is the harmonic frequency of the magnetic field distribution in the permanent magnet sub-field and the air gap sub-field, namely the harmonic frequency of the magnetic field distribution in the permanent magnet sub-field and the air gap sub-field is the same, the maximum value is represented by K, L K Representing dimension k=k max Line vector L of (1) K ＝(1) 1×K ；G 2 Is a second motor structural parameter matrix, G 2k Is the structural parameter coefficient of the kth second motor, meets G 2 ＝diag(G 21 ,G 22 ,…,G 2K )，G 2k ＝(R m /R s ) k ；I Q An identity matrix with dimension Q; sigma (sigma) q0 Fifth motor structural parameter matrix, sigma, representing the q-th slot sub-field q0 (k) The kth fifth motor structural parameter coefficient representing the qth slot subdomain satisfies sigma q0 ＝diag(σ q0 (1),σ q0 (2),...,σ q0 (K)) K×K ，σ q0 (k)＝2sin(kb oa /2)cos(kα q )/(kb oa )；α q The groove center position angle of the q-th groove sub-field; τ q0 Sixth motor structural parameter matrix, τ, representing a q-th slot sub-field q0 (k) The kth and sixth motor structural parameter coefficients representing the qth slot subdomain satisfy τ q0 ＝diag(τ q0 (1),τ q0 (2),...,τ q0 (K)) K×K ,τ q0 (k)＝2sin(kb oa /2)sin(kα q )/(kb oa )；
Figure GDA0004192900430000145
Figure GDA0004192900430000146
Figure GDA0004192900430000147
Wherein ρ is x1 Representing a fifth flux linkage coefficient matrix satisfying ρ x1 ＝diag(sin(π/2),sin(π),...,sin(Nπ/2))；ρ x4 Represents a tenth flux linkage coefficient matrix satisfying ρ x4 ＝diag(cos(π),cos(2π),...,cos(Nπ))；ζ 0 Representing a seventh motor structural parameter matrix, ζ 0 (n) represents the nth seventh motor structural parameter coefficient satisfying ζ 0 ＝diag(ζ 0 (1),ζ 0 (2),...,ζ 0 (N)) N×N ,ζ 0 (n)＝2cos(nπ/2)sin(E n b oa /2)/(nπγ a )，γ a Is the ratio of the width of the groove opening to the width of the groove, and satisfies gamma a ＝b oa /b sa ；E n A harmonic matrix distributed for the slot subdomain magnetic field, satisfies E n ＝diag(E 1 ,E 2 ,...,E N )，E n The harmonic frequency of the magnetic field distribution of the groove subdomain is expressed, and E is satisfied n ＝nπ/b sa N represents the harmonic frequency coefficient of the slot subfield magnetic field distribution, n=1, 2,3 …, the maximum of which is represented by N;L N representing the dimension n=n max Line vector L of (1) N ＝(1) 1×N ；I N For dimension n=n max Is a matrix of units of (a); g 3 Is a third motor structure parameter matrix, meets G 3 ＝diag(G 31 ,G 32 ,…,G 3N )，G 3n Representing the structural parameter coefficient of the nth third motor, G 3n ＝(R t /R sb ) En ；N＝diag(1,2,…,N)。
For the double layer concentrated winding shown in fig. 2, the correlation matrix C in step 2.3) x The following formula is satisfied:
Figure GDA0004192900430000151
wherein s is aq,x ，s bq,x Sum s cq,x Respectively represent the reference directions of a, b and c phase currents in the x-th layer winding in the Q-th slot, q=1, 2, …, Q, x=i, II and s aq,x ，s bq,x Sum s cq,x The value of 1 indicates that current is flowing, -1 indicates that current is flowing, and 0 indicates that there is no phase current.
3) According to the coupling relation of the motor line voltage vector, the phase current vector and the stator winding flux linkage vector and the coupling relation of the motor phase current vector, the stator winding flux linkage vector and the motor structural parameter, establishing an analytic relation of the motor line voltage vector, the phase current vector and the motor structural parameter by taking the stator winding flux linkage vector as an intermediate quantity;
the analytic relation of the motor line voltage vector, the phase current vector and the motor structural parameters meets the following formula:
K 103 A 2 +K 104 B 2 +K 105 C 2 +K 106 D 2 +K 109 D 3t +K 1010 i ab ＝Y 10
wherein,,
K 103 ＝ρ c1 ρ c3 C + (C I +C II )ρ x6 /N a ；
K 104 ＝ρ c1 ρ c3 C + (C I +C II )ρ x6 G 2 /N a ；
K 105 ＝ρ c1 ρ c3 C + (C I +C II )ρ x7 /N a ；
K 106 ＝ρ c1 ρ c3 C + (C I +C II )ρ x7 G 2 /N a ；
K 109 ＝C + [C I (ρ x2s -ρ x3s )-C II (ρ x2s +ρ x3s )]/N a ；
Figure GDA0004192900430000152
Figure GDA0004192900430000153
K 103 a first coefficient matrix, K, representing a variant of the voltage balance equation 104 A second coefficient matrix, K, representing a variant of the voltage balance equation 105 A third coefficient matrix, K, representing a variant of the voltage balance equation 106 A fourth coefficient matrix, K, representing a variant of the voltage balance equation 109 A fifth coefficient matrix, K, representing a variant of the voltage balance equation 1010 A sixth coefficient matrix representing a variant of the voltage balance equation, Y 10 A seventh coefficient matrix representing a variation of the voltage balance equation.
4) In order to realize simultaneous solving of vector magnetic potential and phase current, utilizing interface boundary conditions of continuous magnetic density radial components and continuous magnetic field strength tangential components between adjacent subfields, establishing constraint equations satisfied by first-ninth undetermined coefficients and phase current in vector magnetic potential flux solution expressions of all subfields, and constructing a lumped solving matrix by combining analytic relational expressions of motor line voltage vectors, phase current vectors and motor structural parameters in the step 3);
the step 4) is specifically as follows:
according to the boundary conditions of the interface of the notch subdomain and the slot subdomain, obtaining:
K 87 C 4t +K 88 D 4t +K 89 D 3t +K 810 i ab ＝0
K 97 C 4t +K 98 D 4t +K 99 D 3t +K 910 i ab ＝0
wherein C is 4t For the seventh coefficient vector to be determined, C 4t Is formed by C 4qm Component column vectors, C 4t ＝[C 41 ,C 42 ,…,C 4Q ] T ，C 4q ＝[C 4q1 ,C 4q2 ,…,C 4qM ] T ，D 4t For the eighth coefficient vector to be determined, D 4t Is formed by D 4qm Component column vectors, D 4t ＝[D 41 ,D 42 ,…,D 4Q ] T ，D 4q ＝[D 4q1 ,D 4q2 ,…,D 3qM ] T ；K 87 -K 910 A second nineteenth-thirty-sixth coefficient matrix for the lumped solution matrix of the field coupling resolution model,
Figure GDA0004192900430000161
Figure GDA0004192900430000162
the seventh boundary condition coefficient matrix for the q-th slot subdomain satisfies
Figure GDA0004192900430000163
Figure GDA0004192900430000164
Figure GDA0004192900430000165
An eighth boundary condition coefficient matrix for the q-th slot subdomain meets the following conditions
Figure GDA0004192900430000166
Figure GDA0004192900430000167
Figure GDA0004192900430000168
A ninth boundary condition coefficient matrix for the q-th slot subdomain, satisfies
Figure GDA0004192900430000169
Figure GDA00041929004300001610
Figure GDA00041929004300001611
A tenth boundary condition coefficient matrix for the q-th slot subfield satisfying +.>
Figure GDA00041929004300001612
Figure GDA00041929004300001613
Figure GDA00041929004300001614
An eleventh boundary condition coefficient matrix for the q-th slot subfield satisfying +.>
Figure GDA00041929004300001615
Figure GDA00041929004300001616
Figure GDA00041929004300001617
A twelfth boundary condition coefficient matrix for the q-th slot subfield satisfying +.>
Figure GDA00041929004300001618
Figure GDA00041929004300001619
Figure GDA00041929004300001620
K 810_1 For the first matrix of boundary condition coefficients,
Figure GDA00041929004300001621
for the second boundary condition coefficient matrix of the q-th slot sub-field, satisfy +.>
Figure GDA00041929004300001622
K 910_1 For the third boundary condition coefficient matrix, ++>
Figure GDA00041929004300001623
Fourth boundary condition coefficient matrix for the q-th slot subdomain, satisfy +.>
Figure GDA00041929004300001624
K 910_2 For the fifth boundary condition coefficient matrix, +.>
Figure GDA00041929004300001625
A sixth boundary condition coefficient matrix for the q-th slot subdomain, satisfies
Figure GDA00041929004300001626
G 4 For the fourth motor structural parameter matrix, G 4m Is the structural parameter coefficient of the mth fourth motor and meets G 4 ＝diag(G 41 ,G 42 ,…,G 4M )，/>
Figure GDA00041929004300001627
Figure GDA0004192900430000171
Figure GDA0004192900430000172
Figure GDA0004192900430000173
Wherein, gamma 0 Representing a twelfth motor structural parameter matrix, gamma 0 (n) represents the nth twelfth motor structural parameter coefficient satisfying γ 0 ＝(γ 0 (n)) 1×N The method comprises the steps of carrying out a first treatment on the surface of the ζ is a thirteenth motor structural parameter matrix, satisfying ζ=γ/γ a Gamma represents a fourteenth motor structural parameter matrix, gamma (m, n) represents a fourteenth motor structural parameter coefficient when the harmonic order coefficient of the slot sub-field distribution is m and the harmonic order coefficient of the slot sub-field distribution is n, satisfying gamma= (gamma (m, n)) M×N ；
And obtaining according to boundary conditions of interfaces of the notch subdomain and the air gap subdomain:
K 53 A 2 +K 54 B 2 +K 57 C 4t +K 58 D 4t +K 510 i ab ＝0
K 65 C 2 +K 66 D 2 +K 67 C 4t +K 68 D 4t +K 610 i ab ＝0
K 73 A 2 +K 74 B 2 +K 75 C 2 +K 76 D 2 +K 77 C 4t +K 78 D 4t ＝0
wherein K is 53 -K 78 Thirteenth-twenty-eighth coefficient matrices of the matrix are solved for lumped of the field coupling analytical model.
K 53 ＝K 65 ＝-K
K 54 ＝K 66 ＝G 1 K
K 57 ＝η T F tm G 4t
K 58 ＝-η T F tm
K 67 ＝ξ T F tm G 4t
K 68 ＝-ξ T F tm
Figure GDA0004192900430000174
/>
Figure GDA0004192900430000175
K 73 ＝σ
K 74 ＝σG 2
K 75 ＝τ
K 76 ＝τG 2
K 77 ＝-G 4t
K 78 ＝-I MQ
G 1 For the first motor structural parameter matrix, G 1k Is the structural parameter coefficient of the kth first motor, meets G 1 ＝diag(G 11 ,G 12 ,…,G 1K )，G 1k ＝(R r /R m ) k The method comprises the steps of carrying out a first treatment on the surface of the Eta is a fifteenth motor structural parameter matrix, and meets the following requirements
Figure GDA0004192900430000181
η q Sixteenth motor structural parameter matrix, eta q (m, k) is the structural parameter coefficient of the sixteenth motor, and satisfies eta q ＝(η q (m,k)) M×K ，
Figure GDA0004192900430000182
ζ is the seventeenth motor structure parameter matrix, which satisfies +.>
Figure GDA0004192900430000183
ξ q Eighteenth motor structural parameter matrix, ζ q (m, k) is the structural parameter coefficient of the eighteenth motor, satisfies the requirement of xi q ＝(ξ q (m,k)) M×K ，/>
Figure GDA0004192900430000184
Sigma and tau are nineteenth and twentieth motor structural parameter matrixes respectively, and the sigma=2pi eta/b is satisfied oa And τ=2ζ/b oa ；F tm For the total matrix of harmonic times of notch subdomain magnetic field distribution, F m The harmonic frequency matrix distributed for the notch subdomain magnetic field satisfies F tm ＝diag(F m ,F m ,...,F m ) Q×Q ，F m ＝diag(F 1 ,F 2 ,...,F M )；G 4t Is the total matrix of the structural parameters of the fourth motor, and meets G 4t ＝diag(G 4 ,G 4 ,...,G 4 ) Q×Q ；I MQ Is an M-Q dimensional identity matrix; η (eta) 0 Represents an eighth motor structural parameter matrix, satisfies the following conditions
Figure GDA0004192900430000185
η q0 Represents a ninth motor structural parameter matrix, eta q0 (k) Represents the structural parameter coefficient of the kth and ninth motors, and satisfies eta q0 ＝(η q0 (k)) 1×K ；ξ 0 Represents a tenth motor structural parameter matrix, satisfies +.>
Figure GDA0004192900430000186
ξ q0 Representing an eleventh motor structural parameter matrix, xi q0 (k) Represents the structural parameter coefficient of the kth and eleventh motor, satisfies the requirement of xi q0 ＝(ξ q0 (k)) 1×K ；
γ 0 (n)＝4cos(nπ/2)sin(E n b oa /2)/nπ；
Figure GDA0004192900430000187
Figure GDA0004192900430000188
Figure GDA0004192900430000189
Wherein alpha is q The central position angle of each groove; f (F) m The harmonic frequency of the notch subdomain magnetic field distribution is satisfied with F m ＝mπ/b oa M represents the harmonic frequency coefficient of the notch subfield magnetic field distribution, satisfying m=1, 2,3 …, the maximum value of which is represented by M;
according to the boundary conditions of the interfaces of the permanent magnet subdomains and the air gap subdomains, obtaining:
K 11 A 1 +K 13 A 2 +K 14 B 2 ＝Y 1
K 22 C 1 +K 25 C 2 +K 26 D 2 ＝Y 2
K 31 A 1 +K 33 A 2 +K 34 B 2 ＝Y 3
K 42 C 1 +K 45 C 2 +K 46 D 2 ＝Y 4
K 11 -K 46 a first-twelfth coefficient matrix of the lumped solution matrix for the field coupling analytical model; y is Y 1 -Y 4 A seventeenth-forty coefficient matrix of the lumped solution matrix of the field path coupling analytic model;
K 11 ＝K 22 ＝I K +G 1 2 ；
K 13 ＝K 25 ＝-G 1 2 ；
K 14 ＝K 26 ＝-I K ；
K 31 ＝K 42 ＝I K -G 1 2 ；
K 33 ＝K 45 ＝-μ r G 2 ；
K 34 ＝K 46 ＝μ r I K ；
Y 1 ＝-μ 0 (K 2 -I K ) -1 [(R r KG 1 +R m I K )M αck -(R r G 1 +R m K)M rsk ]；
Y 2 ＝-μ 0 (K 2 -I K ) -1 [(R r KG 1 +R m I K )M αsk +(R r G 1 +R m K)M rck ]；
Y 3 ＝-μ 0 (K 2 -I K ) -1 [K(R m I K -R r G 1 )M αck -(R m I K -R r G 1 )M rsk ]；
Y 4 ＝-μ 0 (K 2 -I K ) -1 [K(R m I K -R r G 1 )M αsk +(R m I K -R r G 1 )M rck ]；
μ r is the relative permeability of the permanent magnet; i K For dimension k=k max Is a matrix of units of (a); a is that 1 And C 1 The first coefficient vector and the second coefficient vector to be determined in the general solution expression of the vector magnetic bits respectively satisfy A 1 ＝[A 11 ,A 12 ,…,A 1K ] T Is composed of undetermined coefficient A 1k Component column vectors, C 1 Satisfy C 1 ＝[C 11 ,C 12 ,…,C 1K ] T Is composed of undetermined coefficient C 1k Component column vectors, M rck And M rsk Respectively M r A cosine coefficient vector and a sine coefficient vector after Fourier decomposition; m is M αck 、M αsk Respectively M α A cosine coefficient vector and a sine coefficient vector after Fourier decomposition; k=diag (1, 2, …, K);
M rck ＝[M rc1 ,M rc1 ,…,M rcK ]，M rsk 、M αck 、M αsk and M is as follows rck In the same form, for parallel charged permanent magnets there are
Figure GDA0004192900430000191
Wherein omega r Is the mechanical angular velocity of the rotor, t is the time of rotation of the rotor, alpha 0 For the initial position angle of the rotor,
Figure GDA0004192900430000192
wherein the first coefficient S of the magnetizing function 1k And a second coefficient S 2k Expressed as:
Figure GDA0004192900430000201
Figure GDA0004192900430000202
B r the residual magnetic density of the permanent magnet is p is the pole pair number of the motor, alpha p Is the polar arc coefficient.
The lumped solution matrix of the field coupling analytical model is obtained by the method:
Figure GDA0004192900430000203
5) FIG. 3 is a block diagram of a field coupling analysis model, in which the time steps are discrete sampling intervals within each time step, based on the motor line voltage vector u for the current time step L Phase current vector i for the last time step ab # Stator winding flux linkage vector ψ # And structural parameters of the motor, solving a lumped solving matrix, and simultaneously obtaining a phase current vector i of the current time step ab And first-ninth coefficient vectors A to be determined in general solution expressions of magnetic bits of vectors of all subfields 1 、C 1 、A 2 、B 2 、C 2 、D 2 、C 4t 、D 4t D (D) 3t The vector magnetic potential and the phase current of each subdomain are solved simultaneously. According to step 2)The coupling relation among the phase current vector, the stator winding flux linkage vector and the motor structural parameters of the motor is calculated to obtain the stator winding flux linkage vector psi of the current time step, and the phase current vector i of the current time step is stored ab And the stator winding flux linkage vector psi of the current time step is used for calculating the next time step;
6) According to the general solution expression of the vector magnetic bits of the air gap sub-fields in each sub-field, the radial component and the tangential component of the magnetic density of the air gap sub-field are calculated, and then the output torque and the torque fluctuation rate of the motor are analyzed, specifically, the radial component and the tangential component of the magnetic density of each sub-field can be calculated according to the general solution expression of the vector magnetic bits of each sub-field, and the calculation formula of the output torque is as follows:
Figure GDA0004192900430000204
wherein T is z To output torque, B 2r And B 2α Respectively representing the magnetic density radial component and tangential component of an air gap subdomain, wherein ∈represents integration, and R is the radial position of a certain point in space under a polar coordinate system, so that R is satisfied m <r<R s When calculating the output torque, r= (R m +R s )/2. From the above equation, the output torque is only related to the radial and tangential components of the flux density of the air gap sub-field, and thus only the calculation results of the radial and tangential components of the flux density of the air gap sub-field are given.
Comparison of simulation results
The test motor is kept to run at the rated rotation speed, and the SPWM modulation mode is adopted to modulate the ratio m m =0.8. When the voltage of the direct current bus is 470V and the power factor is 0.95, the motor meets the rated operation condition. At carrier frequency f c Time step t =5 kHz s Under the condition of =1μs, fig. 4 is a phase current waveform calculated by providing a model, fig. 5 is a phase winding flux linkage waveform, fig. 6 (a) and fig. 6 (b) are radial component and tangential component waveforms of air gap flux density, respectively, and fig. 7 is a permanent magnet motor output torque waveform. Meanwhile, for comparison, FIGS. 4-7 show the fields under the same simulation conditionsAs can be seen from the graph, the calculation results of the two models are very close. Meanwhile, through analysis and calculation, the output torque fluctuation rate obtained by the simulation analysis of the field coupling analysis model and the field coupling time-step finite element is 8.62% and 8.47%, respectively, and the fact that the field coupling analysis model can realize the accurate analysis of the output torque and the torque fluctuation rate of the permanent magnet motor under the excitation of pulse voltage is proved.
The specific implementation of the method of the present invention will be described in detail herein, centering on the embodiments of the present invention. It should be understood that the description herein is merely for the purpose of describing the present invention with respect to the motor structure of the given embodiments, and that the analysis of the performance parameters of the two-layer concentrated winding surface-mounted permanent magnet motor system of different structures may vary in certain details, and such variations are within the scope of the present invention.

Claims (6)

1. The field path coupling analysis method of the surface-mounted permanent magnet motor driving system based on the analysis method is characterized by comprising the following steps of:
1) For a permanent magnet motor with three-phase windings in a star connection mode, establishing a coupling relation of a motor line voltage vector, a phase current vector and a stator winding flux linkage vector according to a voltage balance equation of the permanent magnet motor;
2) Establishing a coupling relation among a phase current vector of the permanent magnet motor, a stator winding flux linkage vector and motor structural parameters by a magnetic field analysis calculation method;
3) Establishing an analytic relational expression of the motor line voltage vector, the phase current vector and the motor structural parameter according to the coupling relation of the motor line voltage vector, the phase current vector and the stator winding flux linkage vector and the coupling relation of the phase current vector, the stator winding flux linkage vector and the motor structural parameter of the permanent magnet motor;
4) Establishing constraint equations satisfied by first-ninth undetermined coefficients and phase currents in the magnetic flux-position solution expressions of all sub-field vectors by utilizing interface boundary conditions of continuous magnetic density radial components and continuous magnetic field intensity tangential components between all adjacent sub-fields, and constructing a lumped solution matrix by combining analytic relational expressions of motor line voltage vectors, phase current vectors and motor structural parameters in the step 3);
5) In each time step, according to the motor line voltage vector of the current time step, the phase current vector of the previous time step, the stator winding flux linkage vector and the structural parameters of the motor, solving a lumped solving matrix, simultaneously obtaining the phase current vector of the current time step and the first-ninth undetermined coefficient vector in the general solution expression of the flux position of each sub-field vector, according to the coupling relation of the phase current vector of the motor, the stator winding flux linkage vector and the structural parameters of the motor in the step 2), calculating to obtain the stator winding flux linkage vector of the current time step, and storing the phase current vector of the current time step and the stator winding flux linkage vector of the current time step for calculating the next time step;
6) And calculating the radial component and tangential component of the magnetic density of the air gap subdomain according to the general solution expression of the vector magnetic bits of the air gap subdomain in each subdomain, and further analyzing the output torque and the torque fluctuation rate of the motor.
2. The analytical method-based field coupling analysis method for the surface-mounted permanent magnet motor driving system according to claim 1, wherein the coupling relation among the motor line voltage vector, the phase current vector and the stator winding flux linkage vector in the step 1) satisfies the following formula:
Figure QLYQS_1
wherein t is s Is the time step; u (u) L ＝[u ab u bc ] T Line voltage vector representing current time step, u ab And u bc Line voltages between a phase and b phase and between b phase and c phase respectively; i.e ab And psi respectively represents a phase current vector of the current time step and a flux linkage vector of the stator winding, thereby meeting i ab ＝[i a i b ] T ，i a ，i b Respectively representing a phase current and a phase current of b phase current; ψ= [ ψ ] a ψ b ψ c ] T ，ψ a ，ψ b ，ψ c Respectively a, b and c phase winding flux linkage, wherein T represents matrix transposition operation; i.e ab # Psi (phi) # Phase current vectors and stator winding flux linkage vectors respectively representing the previous time step; r is phase resistance, L 0 Is leakage inductance; c (C) + A first coefficient matrix representing a voltage balance equation, satisfying
Figure QLYQS_2
C-second coefficient matrix representing the voltage balance equation, satisfying +.>
Figure QLYQS_3
3. The analytical method-based field coupling analysis method for the surface-mounted permanent magnet motor driving system according to claim 2, wherein the step 2) is specifically as follows:
2.1 Simplifying a magnetic field solving model of the motor into a two-dimensional model, and dividing a magnetic field solving area of the two-dimensional model into areas to obtain all the subareas, wherein each subarea comprises a permanent magnet subarea, an air gap subarea, a slot subarea and a slot subarea;
2.2 Solving partial differential equations satisfied by the vector magnetic bits of all the subfields by using a separation variable method to obtain a general solution expression of the vector magnetic bit equation of all the subfields, and further obtaining flux linkage of each layer of winding turn chain in the slot subfields after integrating the vector magnetic bits of the slot subfields;
2.3 Presetting an incidence matrix of stator windings in Q slot subdomains, obtaining an expression of a stator winding flux linkage vector according to flux linkage of each layer of winding turn linkage in each slot and a corresponding incidence matrix, and using the expression as a coupling relation of a phase current vector of a motor, the stator winding flux linkage vector and motor structural parameters, wherein the coupling relation satisfies the following formula:
Figure QLYQS_4
wherein A is 2 For the third coefficient vector to be determined, B 2 For the fourth coefficient vector to be determined, C 2 For the fifth coefficient vector to be determined, D 2 For the sixth coefficient vector to be determined, D 3t Is a ninth undetermined coefficient vector; ρ c1 Representing the first flux linkage coefficient to satisfy ρ c1 ＝lN c b sa /S h ，ρ c2 Representing the second flux linkage coefficient, satisfying ρ c2 ＝μ 0 N c /N a /S h ，ρ c3 Representing the third flux linkage coefficient, satisfying ρ c3 ＝(R sb 2 -R t 2 )/4，ρ c4 Represents the fourth flux linkage coefficient, satisfies
Figure QLYQS_5
Q is the number of stator slots, b oa For wide angle of notch b sa For the stator slot width angle, N c For the number of turns of each coil S h For the area occupied by each layer of winding in the slot, N a The number of parallel branches is l, the effective length of the motor is mu 0 Is air permeability, R r 、R m 、R s 、R t 、R sb Respectively representing the outer radius of the rotor, the outer radius of the permanent magnet, the inner radius of the stator, the arc radius of the groove top and the arc radius of the groove bottom; ρ x2s Representing a sixth flux linkage coefficient matrix,>
Figure QLYQS_6
a seventh flux linkage coefficient matrix representing a sub-field of the q-th slot, satisfying +.>
Figure QLYQS_7
diag () represents a diagonal matrix; ρ x3s Represents an eighth flux linkage coefficient matrix,>
Figure QLYQS_8
a ninth flux linkage coefficient matrix representing a q-th slot subfield satisfying
Figure QLYQS_9
ρ x5s Representing an eleventh flux linkage coefficient matrix,>
Figure QLYQS_10
a twelfth flux linkage coefficient matrix representing a q-th slot subfield satisfying +.>
Figure QLYQS_11
ρ x6 Representing a thirteenth flux linkage coefficient matrix satisfying ρ x6 ＝[(L K σ 10 ) T ,(L K σ 20 ) T ,...,(L K σ Q0 ) T ] T Q×K ，ρ x7 Represents a fourteenth flux linkage coefficient matrix satisfying ρ x7 ＝[(L K τ 10 ) T ,(L K τ 20 ) T ,...,(L K τ Q0 ) T ] T Q×K K is the harmonic frequency of the magnetic field distribution in the permanent magnet sub-field and the air gap sub-field, the maximum value of which is represented by K, L K Line vector representing dimension K, L K ＝(1) 1×K ；G 2 Is a second motor structural parameter matrix, G 2k Is the structural parameter coefficient of the kth second motor, meets G 2 ＝diag(G 21 ,G 22 ,…,G 2K )，G 2k ＝(R m /R s ) k ；I Q An identity matrix with dimension Q; sigma (sigma) q0 Fifth motor structural parameter matrix, sigma, representing the q-th slot sub-field q0 (k) The kth fifth motor structural parameter coefficient representing the qth slot subdomain satisfies sigma q0 ＝diag(σ q0 (1),σ q0 (2),...,σ q0 (K)) K×K ，σ q0 (k)＝2sin(kb oa /2)cos(ka q )/(kb oa )；α q The groove center position angle of the q-th groove sub-field; τ q0 Sixth motor structural parameter matrix, τ, representing a q-th slot sub-field q0 (k) The kth and sixth motor structural parameter coefficients representing the qth slot subdomain satisfy τ q0 ＝diag(τ q0 (1),τ q0 (2),...,τ q0 (K)) K×K ,τ q0 (k)＝2sin(kb oa /2)sin(ka q )/(kb oa )；
Figure QLYQS_12
Figure QLYQS_13
Figure QLYQS_14
Wherein ρ is x1 Representing a fifth flux linkage coefficient matrix satisfying ρ x1 ＝diag(sin(π/2),sin(π),...,sin(Nπ/2))；ρ x4 Represents a tenth flux linkage coefficient matrix satisfying ρ x4 ＝diag(cos(π),cos(2π),...,cos(Nπ))；ζ 0 Representing a seventh motor structural parameter matrix, ζ 0 (n) represents the nth seventh motor structural parameter coefficient satisfying ζ 0 ＝diag(ζ 0 (1),ζ 0 (2),...,ζ 0 (N)) N×N ,ζ 0 (n)＝2cos(nπ/2)sin(E n b oa /2)/(nπγ a )，γ a Is the ratio of the width of the groove opening to the width of the groove, and satisfies gamma a ＝b oa /b sa ；E n A harmonic matrix distributed for the slot subdomain magnetic field, satisfies E n ＝diag(E 1 ,E 2 ,...,E N )，E n The harmonic frequency of the magnetic field distribution of the groove subdomain is expressed, and E is satisfied n ＝nπ/b sa N represents the harmonic frequency coefficient of the slot subfield magnetic field distribution, n=1, 2,3 …, the maximum of which is represented by N; l (L) N Representing a row vector of dimension N, L N ＝(1) 1×N ；I N An identity matrix with dimension N; g 3 Is a third motor structure parameter matrix, meets G 3 ＝diag(G 31 ,G 32 ,…,G 3N )，G 3n Representing the structural parameter coefficient of the nth third motor, G 3n ＝(R t /R sb ) En ；N＝diag(1,2,…,N)；
The correlation matrix C in the step 2.3) x The following formula is satisfied:
Figure QLYQS_15
wherein s is aq,x ，s bq,x Sum s cq,x Respectively represent the reference directions of a, b and c phase currents in the x-th layer winding in the Q-th slot, q=1, 2, …, Q, x=i, II and s aq,x ，s bq,x Sum s cq,x The value of 1 indicates that current is flowing, -1 indicates that current is flowing, and 0 indicates that there is no phase current.
4. The analytical method-based field path coupling analysis method of the surface-mounted permanent magnet motor driving system according to claim 3, wherein the condition for simplifying the electromagnetic field of the motor in the step 2.1) is specifically as follows:
the magnetic permeability of stator and rotor iron cores is infinite; the demagnetizing curve of the permanent magnetic material is linear; thirdly, ignoring the conductivity effect and the eddy current effect of the ferromagnetic material; the vector magnetic bit of each subdomain only contains axial components; and fifthly, simplifying the structures of the grooves and the notches into fan-shaped structures.
5. The analytical method-based field coupling analysis method of the surface-mounted permanent magnet motor driving system according to claim 3, wherein the analytical relation among the motor line voltage vector, the phase current vector and the motor structural parameters in the step 3) satisfies the following formula:
K 103 A 2 +K 104 B 2 +K 105 C 2 +K 106 D 2 +K 109 D 3t +K 1010 i ab ＝Y 10
wherein,,
K 103 ＝ρ c1 ρ c3 C + (C I +C II )ρ x6 /N a ；
K 104 ＝ρ c1 ρ c3 C + (C I +C II )ρ x6 G 2 /N a ；
K 105 ＝ρ c1 ρ c3 C + (C I +C II )ρ x7 /N a ；
K 106 ＝ρ c1 ρ c3 C + (C I +C II )ρ x7 G 2 /N a ；
K 109 ＝C + [C I (ρ x2s -ρ x3s )-C II (ρ x2s +ρ x3s )]/N a ；
Figure QLYQS_16
Figure QLYQS_17
K 103 a first coefficient matrix, K, representing a variant of the voltage balance equation 104 A second coefficient matrix, K, representing a variant of the voltage balance equation 105 A third coefficient matrix, K, representing a variant of the voltage balance equation 106 A fourth coefficient matrix, K, representing a variant of the voltage balance equation 109 A fifth coefficient matrix, K, representing a variant of the voltage balance equation 1010 A sixth coefficient matrix representing a variant of the voltage balance equation, Y 10 A seventh coefficient matrix representing a variation of the voltage balance equation.
6. The analytical method-based field coupling analysis method for the surface-mounted permanent magnet motor driving system according to claim 3, wherein the step 4) is specifically:
according to the boundary conditions of the interface of the notch subdomain and the slot subdomain, obtaining:
K 87 C 4t +K 88 D 4t +K 89 D 3t +K 810 i ab ＝0
K 97 C 4t +K 98 D 4t +K 99 D 3t +K 910 i ab ＝0
and obtaining according to boundary conditions of interfaces of the notch subdomain and the air gap subdomain:
K 53 A 2 +K 54 B 2 +K 57 C 4t +K 58 D 4t +K 510 i ab ＝0
K 65 C 2 +K 66 D 2 +K 67 C 4t +K 68 D 4t +K 610 i ab ＝0
K 73 A 2 +K 74 B 2 +K 75 C 2 +K 76 D 2 +K 77 C 4t +K 78 D 4t ＝0
according to the boundary conditions of the interfaces of the permanent magnet subdomains and the air gap subdomains, obtaining:
K 11 A 1 +K 13 A 2 +K 14 B 2 ＝Y 1
K 22 C 1 +K 25 C 2 +K 26 D 2 ＝Y 2
K 31 A 1 +K 33 A 2 +K 34 B 2 ＝Y 3
K 42 C 1 +K 45 C 2 +K 46 D 2 ＝Y 4
wherein A is 1 For the first coefficient vector to be determined, C 1 For the second coefficient vector to be determined, C 4t For the seventh coefficient vector to be determined, D 4t Is an eighth undetermined coefficient vector; k (K) 11 、K 13 、K 14 、K 22 、K 25 、K 26 、K 31 、K 33 、K 34 、K 42 、K 45 、K 46 、K 53 、K 54 、K 57 、K 58 、K 65 、K 66 、K 67 、K 68 、K 73 、K 74 、K 75 、K 76 、K 77 、K 78 、K 87 、K 88 、K 89 、K 97 、K 98 、K 99 、K 510 、K 610 、K 810 、K 910 Y is as follows 1 -Y 4 First-fourth coefficient matrixes of the lumped solving matrix of the field path coupling analytic model are respectively;
Figure QLYQS_18
Figure QLYQS_19
Figure QLYQS_20
Figure QLYQS_21
wherein K is 810_1 For the first matrix of boundary condition coefficients,
Figure QLYQS_24
for the second boundary condition coefficient matrix of the q-th slot sub-field, satisfy +.>
Figure QLYQS_25
K 910_1 For the third boundary condition coefficient matrix, ++>
Figure QLYQS_27
Fourth boundary condition coefficient matrix for the q-th slot subdomain, satisfy +.>
Figure QLYQS_23
K 910_2 For the fifth boundary condition coefficient matrix, +.>
Figure QLYQS_26
A sixth boundary condition coefficient matrix for the q-th slot subdomain, satisfies
Figure QLYQS_28
η 0 Represents an eighth motor structural parameter matrix, satisfies the following conditions
Figure QLYQS_29
η q0 Represents a ninth motor structural parameter matrix, eta q0 (k) Represents the structural parameter coefficient of the kth and ninth motors, and satisfies eta q0 ＝(η q0 (k)) 1×K ；ξ 0 Represents a tenth motor structural parameter matrix, satisfies +.>
Figure QLYQS_22
ξ q0 Representing an eleventh motor structural parameter matrix, xi q0 (k) Represents the structural parameter coefficient of the kth and eleventh motor, satisfies the requirement of xi q0 ＝(ξ q0 (k)) 1×K ；
Figure QLYQS_30
Figure QLYQS_31
Figure QLYQS_32
/>
Wherein, gamma 0 Representing a twelfth motor structural parameter matrix, gamma 0 (n) represents the nth twelfth motor structural parameter coefficient satisfying γ 0 ＝(γ 0 (n)) 1×N The method comprises the steps of carrying out a first treatment on the surface of the ζ is a thirteenth motor structural parameter matrix, satisfying ζ=γ/γ a Gamma represents a fourteenth motor structural parameter matrix, gamma (m, n) represents a fourteenth motor structural parameter coefficient when the harmonic order coefficient of the slot sub-field distribution is m and the harmonic order coefficient of the slot sub-field distribution is n, satisfying gamma= (gamma (m, n)) M×N ；
γ 0 (n)＝4cos(nπ/2)sin(E n b oa /2)/nπ；
Figure QLYQS_33
Figure QLYQS_34
Figure QLYQS_35
Wherein alpha is q The central position angle of each groove; f (F) m The harmonic frequency of the notch subdomain magnetic field distribution is satisfied with F m ＝mπ/b oa M represents the harmonic frequency coefficient of the notch subfield magnetic field distribution, satisfying m=1, 2,3 …, the maximum value of which is represented by M;
the lumped solution matrix of the field coupling analytical model is obtained by the method:
Figure QLYQS_36
/>
CN202111288041.1A 2021-11-02 2021-11-02 Surface-mounted permanent magnet motor driving system field path coupling analysis method based on analysis method Active CN113992092B (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN202111288041.1A CN113992092B (en) 2021-11-02 2021-11-02 Surface-mounted permanent magnet motor driving system field path coupling analysis method based on analysis method

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN202111288041.1A CN113992092B (en) 2021-11-02 2021-11-02 Surface-mounted permanent magnet motor driving system field path coupling analysis method based on analysis method

Publications (2)

Publication Number Publication Date
CN113992092A CN113992092A (en) 2022-01-28
CN113992092B true CN113992092B (en) 2023-06-02

Family

ID=79745777

Family Applications (1)

Application Number Title Priority Date Filing Date
CN202111288041.1A Active CN113992092B (en) 2021-11-02 2021-11-02 Surface-mounted permanent magnet motor driving system field path coupling analysis method based on analysis method

Country Status (1)

Country Link
CN (1) CN113992092B (en)

Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN107608934A (en) * 2017-08-27 2018-01-19 浙江同星制冷有限公司 A kind of motor radial direction electric and magnetic oscillation Coupled field and circuit analysis method
CN109600006A (en) * 2018-11-30 2019-04-09 浙江大学 A kind of method for solving for surface-mounted permanent magnet machine electromagnetic design
CN111654165A (en) * 2020-05-31 2020-09-11 天津工业大学 Built-in permanent magnet synchronous motor magnetic field calculation method based on rotor magnetic potential model
CN112737173A (en) * 2020-12-17 2021-04-30 天津工业大学 Magnetic field calculation method for segmented oblique pole surface-mounted permanent magnet motor

Family Cites Families (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
JP2007206943A (en) * 2006-02-01 2007-08-16 Sharp Corp Electromagnetic field circuit coupled analysis program, recording medium, analysis method, and analyzer
CN107017666B (en) * 2017-04-06 2019-09-10 中国人民解放军后勤工程学院 Small-sized single-phase synchronous generator paired running system finite element model method for building up
CN107976483B (en) * 2017-11-30 2021-11-19 石家庄铁道大学 Electromagnetic flaw detector for detecting interface cracks of piezoelectric-magnetic laminated structure and detection method
JP6991061B2 (en) * 2017-12-28 2022-01-12 株式会社日立製作所 Power supply voltage waveform calculation method, circuit coupled magnetic field analysis method, program and recording medium on which the program is recorded
CN111651911B (en) * 2020-04-17 2022-09-02 北京理工大学 Lumped element impedance sensitivity rapid calculation method and optimization method
CN112257231B (en) * 2020-09-22 2023-03-21 重庆交通大学 Permanent magnet motor analysis method and system considering nonlinear characteristics of silicon steel sheet and application

Patent Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN107608934A (en) * 2017-08-27 2018-01-19 浙江同星制冷有限公司 A kind of motor radial direction electric and magnetic oscillation Coupled field and circuit analysis method
CN109600006A (en) * 2018-11-30 2019-04-09 浙江大学 A kind of method for solving for surface-mounted permanent magnet machine electromagnetic design
CN111654165A (en) * 2020-05-31 2020-09-11 天津工业大学 Built-in permanent magnet synchronous motor magnetic field calculation method based on rotor magnetic potential model
CN112737173A (en) * 2020-12-17 2021-04-30 天津工业大学 Magnetic field calculation method for segmented oblique pole surface-mounted permanent magnet motor

Also Published As

Publication number Publication date
CN113992092A (en) 2022-01-28

Similar Documents

Publication Publication Date Title
Bianchi et al. Magnetic models of saturated interior permanent magnet motors based on finite element analysis
CN109600006B (en) Solving method for electromagnetic design of surface-mounted permanent magnet motor
Nehl et al. Determination of saturated values of rotating machinery incremental and apparent inductances by an energy perturbation method
Alipour-Sarabi et al. Performance analysis of concentrated wound-rotor resolver for its applications in high pole number permanent magnet motors
Ding et al. A novel mesh-based equivalent magnetic network for performance analysis and optimal design of permanent magnet machines
CN107231111B (en) Direct torque control method for high load capacity of five-phase permanent magnet synchronous motor
CN111651914A (en) Magnetic field analytic calculation method of bearingless flux switching motor
CN113507189B (en) Design parameter optimization method for built-in permanent magnet synchronous motor
Tang et al. Analytical modeling of flux-switching machines using variable global reluctance networks
CN110661468B (en) Method for acquiring electromagnetic field performance of bearingless flux switching motor
CN109768685B (en) Surface-mounted permanent magnet motor armature magnetic field reconstruction method based on single-slot model
CN113343171B (en) Surface-mounted permanent magnet motor magnetic field analysis and calculation method considering stator core saturation
CN113992092B (en) Surface-mounted permanent magnet motor driving system field path coupling analysis method based on analysis method
Cui et al. Stator structure design and analysis of variable reluctance resolver for hybrid-vehicle motor drive
Du et al. Semianalytical model of multiphase Halbach array axial flux permanent-magnet motor considering magnetic saturation
Roggia et al. Two-dimensional analysis for conical permanent magnet motors
Liu et al. Comparative study of rotor PM transverse flux machine and stator PM transverse flux machine with SMC cores
Ding et al. Analytical calculation of no-load air gap magnetic field of surface mounted permanent magnet motor based on Schwarz-Christoffel transformation
Han et al. An effective method of verifying poles polarities in switched reluctance motors
Ruuskanen Design Aspects of Megawatt-Range Direct-Driven Permanent MagnetWind Generators
Babe et al. Analytical Modelling of Doubly Salient Electric Machines using Conformal Mapping Method
CN117973295A (en) Motor equivalent magnetic network model based on magnetic flux source-magnetic resistance-magnetic induction
Mustafa Field Weakening Control of Interior Permanent Magnet Synchronous Motor Employing Model Order Reduction
Zhu et al. Analytical calculation of leakage permeance of coreless axial flux permanent magnet generator
Carbonieri et al. Induction Machine Finite Element Analysis Using Static Simulations: Comparison With The Time-Domain Approach

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
GR01 Patent grant
GR01 Patent grant