Inductance calculation and verification method of fractional-slot concentrated winding permanent magnet synchronous motor

Abstract

The application provides a kind of fractional slot concentrated winding permanent magnet synchronous motor inductance calculation and verification method.First, the winding self-induction is calculated, including armature self-induction, slot self-leakage inductance and tooth top leakage inductance;Then, the winding mutual inductance is calculated, including armature mutual inductance and slot mutual leakage inductance, so, it is completed three-phase inductance calculation;Then, the correctness of three-phase inductance calculation is verified using finite element simulation software;Finally, the dq axis inductance is obtained by coordinate transformation matrix, and the controllability and design rationality of the motor are verified in MATLAB, so the correctness, rationality of motor inductance calculation and controllability verification of the motor are completed.The application solves the problem that the traditional alternating current winding inductance calculation formula is no longer applicable to fractional slot winding inductance calculation, and the calculation process is clear and simple, and the fractional slot concentrated winding inductance can be quickly calculated according to the process.

Description

Technical Field

[0001] This invention belongs to the field of motor design and control technology, specifically relating to a method for calculating and verifying the inductance of a fractional-slot concentrated winding permanent magnet synchronous motor. Background Technology

[0002] Motor manufacturing costs are high, making the initial design and verification of motors extremely important. Winding inductance, as a pivotal parameter in motor design and control, is a crucial factor for constructing accurate mathematical models and analyzing motor performance. It has a significant impact on motor performance, so the correctness of the motor inductance parameter design must be verified in the early stages of the design process to avoid wasting manufacturing costs.

[0003] Fractional slot concentrated winding (FSCW) motors have attracted widespread attention and application due to their short end windings, high efficiency, high power density, good back EMF waveform, and excellent fault tolerance. However, the inductance calculation method for fractional slot winding motors cannot be used as that for traditional integer slot motors. In traditional AC windings, the self-inductance is calculated by adding the fundamental magnetizing inductance and leakage inductance, where the harmonic leakage inductance is very small. While the calculation of the armature reaction fundamental magnetizing inductance for fractional slot concentrated windings is the same as for traditional AC windings, the calculation of the total armature reaction inductance and harmonic leakage inductance differs from that of traditional AC windings because the harmonic leakage inductance is very large, even exceeding the fundamental magnetizing inductance.

[0004] The literature “Chen Yiguang. Inductance Calculation of Fractional Slot Concentrated Winding Permanent Magnet Synchronous Motor [J]. Journal of Electrical Engineering, 2014, 29(3): 119-124” gives the calculation formula for the total armature inductance of FSCW expressed in terms of air gap permeability. The literature “Tong Wenming. Research on Inductance Parameters of Fractional Slot Concentrated Winding Permanent Magnet Synchronous Motor Based on Winding Function Method [J]. Journal of Electrical Engineering, 2015, 30(13): 150-157” gives the calculation formula for the armature inductance of the winding function method expressed in terms of turns function and winding function. The calculation of slot leakage inductance of fractional slots is generally based on the slot pole matching and winding form, giving the leakage inductance coefficients of self-inductance and mutual inductance and the slot ratio permeability, and then calculating the slot leakage inductance.

[0005] However, current research focuses on a portion of the FCSW inductor and lacks a complete overview and comparison of the calculation methods and processes for three-phase inductors. The results are all verified by finite element simulation software, without verifying the rationality of the inductor design or the control effect of the inductor under the control strategy. Summary of the Invention

[0006] To overcome the shortcomings of existing technologies, this invention provides a method for calculating and verifying the inductance of a fractional-slot concentrated winding permanent magnet synchronous motor. First, the winding self-inductance is calculated, including armature self-inductance, slot self-leakage inductance, and tooth tip leakage inductance. Then, the winding mutual inductance is calculated, including armature mutual inductance and slot mutual leakage inductance, thus completing the three-phase inductance calculation. Next, the correctness of the three-phase inductance calculation is verified using finite element simulation software. Finally, the dq-axis inductance is obtained using a coordinate transformation matrix, and the motor's controllability and design rationality are verified using a control strategy in MATLAB. This completes the verification of the correctness and rationality of the motor inductance calculation and the motor's controllability. This invention solves the problem that traditional AC winding inductance calculation formulas are no longer applicable to fractional-slot winding inductance calculations. The calculation process is clear and simple, allowing for rapid calculation of the fractional-slot concentrated winding inductance.

[0007] A method for calculating and verifying the inductance of a fractional-slot concentrated winding permanent magnet synchronous motor, characterized by the following steps:

[0008] Step 1: Calculate the winding self-inductance L according to the following formula. A :

[0009] L A =L AA +L s_self +L t (1)

[0010] Among them, L AA L represents armature self-inductance. s_self Indicates the self-leakage inductance of the groove, L t Indicates leakage at the tooth tip;

[0011] The armature self-inductance L AA The following formula is used to calculate:

[0012]

[0013] Where Z0 represents the number of slots in the unit motor, P0 represents the number of pole pairs in the unit motor, m represents the number of phases in the motor winding, μ0 represents the free permeability, r represents the air gap radius, and l ef N represents the armature lamination length, g represents the equivalent air gap length, and N represents the armature lamination length. c This indicates the number of turns in one coil of the winding;

[0014] The aforementioned slot self-leakage inductance L s_self The following formula is used to calculate:

[0015] L s_self =k self μ0N s 2 l ef λ s (3)

[0016] Where, k self N represents the self-leakage inductance coefficient of the slot. s λ represents the number of conductors per slot. s Indicates the slot ratio leakage permeability;

[0017] The tooth tip leakage inductance L t The following formula is used to calculate:

[0018]

[0019] Where Z represents the total number of slots in the motor, k t λ represents the phase angle difference. t Indicates the tooth ratio leakage permeability;

[0020] The phase angle difference k t The following formula is used to calculate:

[0021]

[0022] Where, α n Let α represent the phase angle difference between the two conductor sides in the nth slot of the motor. If the two conductor sides in the slot are in phase, then α n If the value is 0, then α is not in phase. n It is π / 3;

[0023] The tooth ratio leakage permeability λ t The following formula is used to calculate:

[0024]

[0025] Where b0 represents the width of the slot opening;

[0026] Step 2: Calculate the winding mutual inductance L according to the following formula. Am :

[0027] L Am =L AB +L s_mutual (7)

[0028] Among them, L AB L represents armature mutual inductance. s_mutual Indicates mutual leakage in the slots;

[0029] The armature mutual inductance L AB The following formula is used to calculate:

[0030]

[0031] The aforementioned slot leakage inductance L s_mutual Calculated according to the following formula

[0032] L s_mutual =kmutual μ0N s 2 l ef λ s (9)

[0033] Where, k mutual Indicates the leakage inductance coefficient between slots;

[0034] Step 3: Use finite element simulation software to build a model of the motor. Align the axis of phase A winding with the d-axis. Add current excitation I, -0.5I, and -0.5I to the three-phase windings A, B, and C respectively. Obtain the three-phase inductance values ​​of the motor through electromagnetic field analysis. Compare the three-phase inductance values ​​obtained from the simulation with the three-phase inductance values ​​calculated in steps 1 and 2. If the error between the two is less than 10%, it meets the requirements of motor performance analysis and practical application, proving that the inductance calculation is correct.

[0035] Step 4: Apply the following formula to the three-phase inductor [L] abc The matrix is ​​transformed to obtain the dq-axis inductance:

[0036]

[0037] Among them, L d Inductance is represented by the d-axis coordinate, L q Inductance represented by C on the q-axis lark Represents the Clark transformation matrix. P ark Represents the Park transformation matrix. θ represents the angle between the d-axis and the A-phase axis;

[0038] A motor control system was built using the SIMULINK module in MATLAB. The calculated dq-axis inductance was input, and the correctness of the motor inductance calculation was verified using the SVPWM control strategy. Specifically, the motor speed was set to the rated speed, the motor was accelerated from standstill to the rated speed under no-load, and then the rated load was suddenly applied. The speed, load torque, and phase current waveforms under the dynamic performance of the motor were observed. If the motor operates stably at the rated speed and rated torque, and the phase current amplitude matches the designed rated operating conditions of the motor, it indicates that the motor is controllable and the inductance design meets the actual operating conditions.

[0039] The beneficial effects of this invention are as follows: Considering that the harmonic leakage inductance of the fractional-slot concentrated winding is very large, even greater than the fundamental excitation inductance, the inductance calculation no longer adopts the traditional AC winding inductance calculation method. Based on the armature magnetomotive force waveform, the harmonic leakage inductance is no longer calculated separately. Instead, the winding self-inductance, including armature self-inductance, slot self-leakage inductance, and tooth tip leakage inductance, is calculated first, and then the winding mutual inductance, including armature mutual inductance and slot mutual leakage inductance, is calculated to obtain the three-phase inductance. The correctness of the three-phase inductance calculation is verified by finite element simulation. After obtaining the dq-axis inductance using coordinate transformation, the control strategy is used to verify the consistency between the motor control effect and the design rated operating conditions. This dual verification ensures the correctness of the motor inductance calculation and the feasibility of the design. This invention solves the problem of complex calculation of fractional-slot concentrated winding inductance. The calculation process is clear and simple, and the fractional-slot concentrated winding inductance can be quickly calculated by following the process. Attached Figure Description

[0040] Figure 1 This is a flowchart of the inductance calculation and verification method for a fractional slot concentrated winding permanent magnet synchronous motor according to the present invention.

[0041] Figure 2 This is a schematic diagram of the magnetomotive force of an even-numbered slot phase winding;

[0042] Figure 3 It is a dynamic performance diagram of the motor;

[0043] In the figure, (a) is the speed waveform, (b) is the torque waveform, and (c) is the three-phase current waveform. Detailed Implementation

[0044] The present invention will be further described below with reference to the accompanying drawings and embodiments. The present invention includes, but is not limited to, the following embodiments.

[0045] like Figure 1 As shown, this invention provides a method for calculating and verifying the inductance of a fractional-slot concentrated winding permanent magnet synchronous motor, the specific implementation process of which is as follows:

[0046] Step 1: Calculate the winding self-inductance L according to the following formula. A :

[0047] L A =L AA +L s_self +L t (11)

[0048] Among them, L AA L represents armature self-inductance. s_self Indicates the self-leakage inductance of the groove, L t Indicates leakage at the tooth tip;

[0049] The armature self-inductance L AA The following formula is used to calculate:

[0050]

[0051] Where Z0 represents the number of slots in the unit motor, P0 represents the number of pole pairs in the unit motor, m represents the number of phases in the motor winding, μ0 represents the free permeability, r represents the air gap radius, and l ef N represents the armature lamination length, g represents the equivalent air gap length, and N represents the armature lamination length. c This indicates the number of turns in one coil of the winding;

[0052] The aforementioned slot self-leakage inductance L s_self The following formula is used to calculate:

[0053] L s_self =k self μ0N s 2 l ef λ s (13)

[0054] Where, k self N represents the self-leakage inductance coefficient of the slot. s λ represents the number of conductors per slot. s Indicates the slot ratio leakage permeability;

[0055] The tooth tip leakage inductance L t The following formula is used to calculate:

[0056]

[0057] Where Z represents the total number of slots in the motor, k t λ represents the phase angle difference. t Indicates the tooth ratio leakage permeability;

[0058] The phase angle difference k t The following formula is used to calculate:

[0059]

[0060] Where, α n Let α represent the phase angle difference between the two conductor sides in the nth slot of the motor. If the two conductor sides in the slot are in phase, then α n If the value is 0, then α is not in phase. n It is π / 3;

[0061] The tooth ratio leakage permeability λ t The following formula is used to calculate:

[0062]

[0063] Where b0 represents the width of the slot opening;

[0064] This example uses a 24-slot, 20-pole motor to illustrate the specific calculation method for winding self-inductance. A schematic diagram of the phase winding magnetomotive force of a single-unit motor is shown below. Figure 2 The horizontal axis represents the spatial angle of the motor windings, and the vertical axis represents the magnetomotive force (F) generated by a coil. c This represents the magnetomotive force amplitude. "A+" indicates that phase A current flows into the paper, "A-" indicates that phase A current flows out of the paper, and so on for phases B and C. The unit motor satisfies Z0 = 2P0 ± 2, and the armature self-inductance L... AA for

[0065]

[0066] Slot leakage inductance coefficient k of a 24-slot, 20-pole motor self =6, slot self-leakage inductance L s_self for:

[0067] L s_self =6μ0N s 2 l ef λ s (18)

[0068] The conductor sides in the slots of a 24-slot, 20-pole unit motor are as follows:

[0069] ...|AA-|A+B-|B+B+|BC-|C+C+|CA-|A+A+|A-B+|BB-|B+C-|C+C+|C-A+|...

[0070] The phase angle difference for a 24-slot, 20-pole motor is:

[0071]

[0072] Tooth tip leakage L t for:

[0073]

[0074] Then the winding self-inductance L A :

[0075]

[0076] Step 2: Calculate the winding mutual inductance L according to the following formula. Am :

[0077] L Am =L AB +L s_mutual (twenty two)

[0078] Among them, L AB L represents armature mutual inductance. s_mutual Indicates mutual leakage in the slots;

[0079] The armature mutual inductance L AB The following formula is used to calculate:

[0080]

[0081] The aforementioned slot leakage inductance L s_mutual Calculated according to the following formula

[0082] L s_mutual =k mutual μ0N s 2 l ef λ s (twenty four)

[0083] Where, k mutual Indicates the leakage inductance coefficient between slots;

[0084] Taking a 24-slot, 20-pole motor as an example, the specific calculation method for winding mutual inductance will be explained. The unit motor satisfies Z0 = 2P0 ± 2, and the armature mutual inductance L... AB It is 0.

[0085] Slot leakage inductance coefficient k of a 24-slot, 20-pole motor mutual =-1, slot leakage inductance L s_self for:

[0086] L s_self =-μ0N s 2 l ef λ s (25)

[0087] The mutual inductance of the windings is then:

[0088] L Am =-μ0N s 2 l ef λ s (26)

[0089] Step 3: Use ANSYS software to build a model of the motor, align the axis of phase A winding with the d-axis, and add current excitations I, -0.5I, and -0.5I to the three-phase windings ABC respectively. Obtain the three-phase inductance values ​​of the motor through electromagnetic field analysis. Compare the three-phase inductance values ​​obtained from the simulation with the three-phase inductance values ​​calculated in steps 1 and 2. If the error between the two is less than 10%, it meets the requirements of motor performance analysis and practical application, proving that the inductance calculation is correct.

[0090] Step 4: Apply the following formula to the three-phase inductor [L] abc The matrix is ​​transformed to obtain the dq-axis inductance:

[0091]

[0092] Among them, L d Inductance is represented by the d-axis coordinate, L q Inductance represented by C on the q-axis lark Represents the Clark transformation matrix. P ark Represents the Park transformation matrix. θ represents the angle between the d-axis and the A-phase axis.

[0093] A motor control system was built using the SIMULINK module in MATLAB. The calculated dq-axis inductance was input, and the correctness of the motor inductance calculation was verified using the SVPWM control strategy. Specifically, the motor speed was set to the rated speed, the motor was accelerated from standstill to the rated speed under no-load, and then the rated load was suddenly applied. The speed, load torque, and phase current waveforms under the dynamic performance of the motor were observed. If the motor operates stably at the rated speed and rated torque, and the phase current amplitude matches the designed rated operating conditions of the motor, it indicates that the motor is controllable and the inductance design meets the actual operating conditions.

[0094] This example uses a 24-slot, 20-pole lift-enhancing motor being designed for a distributed all-electric aircraft. The motor's rated operating requirements are: voltage 460VDC, current amplitude 40A, speed 5400rpm, and torque 24Nm. The verification method for the control strategy is explained below.

[0095] The motor speed was set to 5400 rpm. The motor was accelerated from standstill to rated speed under no-load conditions. In 0.4 seconds, the motor suddenly increased from no-load to rated load of 24 Nm. Figure 3 The image shows the dynamic performance waveforms of a motor with a switching frequency of 20 kHz, including speed, load torque, and phase current. The horizontal axis represents time, and the vertical axes represent the speed, torque, and current amplitude, respectively. Figure 3 It can be seen that after a sudden application of the rated load in 0.4 seconds, the speed stabilizes at 5400 rpm, the average rated torque is 24 Nm, and the phase current amplitude is 40 A, all of which are consistent with the designed rated operating conditions of the motor. This indicates that the design of the inductor meets the design requirements, the motor is controllable, and the dynamic performance is good.

Claims

1. A method for calculating and verifying the inductance of a fractional-slot concentrated winding permanent magnet synchronous motor, characterized in that... The steps are as follows: Step 1: Calculate the winding self-inductance L according to the following formula. A : L A L AA +L s_self +L t (1) Among them, L AA L represents armature self-inductance. s_self Indicates the self-leakage inductance of the groove, L t Indicates leakage at the tooth tip; The armature self-inductance L AA The following formula is used to calculate: Where Z0 represents the number of slots in the unit motor, P0 represents the number of pole pairs in the unit motor, m represents the number of phases in the motor winding, μ0 represents the free permeability, r represents the air gap radius, and l ef N represents the armature lamination length, g represents the equivalent air gap length, and N represents the armature lamination length. c This indicates the number of turns in one coil of the winding; The aforementioned slot self-leakage inductance L s_self The following formula is used to calculate: L s_self =k self μ0N s 2 l ef l s (3) Where, k self N represents the self-leakage inductance coefficient of the slot. s λ represents the number of conductors per slot. s Indicates the slot ratio leakage permeability; The tooth tip leakage inductance L t The following formula is used to calculate: Where Z represents the total number of slots in the motor, k t λ represents the phase angle difference. t Indicates the tooth ratio leakage permeability; The phase angle difference k t The following formula is used to calculate: Where, α n Let α represent the phase angle difference between the two conductor sides in the nth slot of the motor. If the two conductor sides in the slot are in phase, then α n If the value is 0, then α is not in phase. n It is π / 3; The tooth ratio leakage permeability λ t The following formula is used to calculate: Where b0 represents the width of the slot opening; Step 2: Calculate the winding mutual inductance L according to the following formula. Am : L Am L AB +L s_mutual (7) Among them, L AB L represents armature mutual inductance. s_mutual Indicates mutual leakage in the slots; The armature mutual inductance L AB The following formula is used to calculate: The aforementioned slot leakage inductance L s_mutual Calculated according to the following formula L s_mutual =k mutual μ0N s 2 l ef l s (9) Where, k mutual Indicates the leakage inductance coefficient between slots; Step 3: Use finite element simulation software to build a model of the motor. Align the axis of phase A winding with the d-axis. Add current excitation I, -0.5I, and -0.5I to the three-phase windings A, B, and C respectively. Obtain the three-phase inductance values ​​of the motor through electromagnetic field analysis. Compare the three-phase inductance values ​​obtained from the simulation with the three-phase inductance values ​​calculated in steps 1 and 2. If the error between the two is less than 10%, it meets the requirements of motor performance analysis and practical application, proving that the inductance calculation is correct. Step 4: Apply the following formula to the three-phase inductor [L] abc The matrix is ​​transformed to obtain the dq-axis inductance: Among them, L d Inductance is represented by the d-axis coordinate, L q Inductance represented by C on the q-axis lark Represents the Clark transformation matrix. P ark Represents the Park transformation matrix. θ represents the angle between the d-axis and the A-phase axis; A motor control system was built using the SIMULINK module in MATLAB. The calculated dq-axis inductance was input, and the correctness of the motor inductance calculation was verified using the SVPWM control strategy. Specifically, the motor speed was set to the rated speed, the motor was accelerated from standstill to the rated speed under no-load, and then the rated load was suddenly applied. The speed, load torque, and phase current waveforms under the dynamic performance of the motor were observed. If the motor operates stably at the rated speed and rated torque, and the phase current amplitude matches the designed rated operating conditions of the motor, it indicates that the motor is controllable and the inductance design meets the actual operating conditions.

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