Vibration suppression method of fractional slot concentrated winding permanent magnet synchronous motor based on radial-tangential electromagnetic force redistribution

Abstract

The invention provides a fractional slot concentrated winding permanent magnet synchronous motor vibration suppression method based on radial-tangential electromagnetic force redistribution. According to the method, a single-side or double-side chamfering structure is introduced into a motor stator tooth portion, air gap flux density distribution is changed, and coordinated modulation of radial and tangential electromagnetic force is achieved. And establishing an equivalent effect model of tangential force on radial vibration by utilizing a force translation theorem, and optimizing air gap electromagnetic force spectrum distribution on the whole. Finite element simulation and experimental verification show that the provided force redistribution strategy can effectively reduce the harmonic amplitude of the electromagnetic force, the vibration response of the motor under typical and atypical harmonic waves is remarkably suppressed, and the effect of the single-side chamfer structure on the aspect of reducing main order vibration is more remarkable. The permanent magnet motor can be widely applied to a high-performance electric drive system and is used for improving the operation stability and noise control performance of the permanent magnet motor.

Description

Technical Field

[0001] This invention relates to the field of vibration and noise control technology for permanent magnet synchronous motors, specifically to a vibration and noise suppression method for fractional-slot concentrated winding permanent magnet synchronous motors based on radial-tangential electromagnetic force redistribution, belonging to the technical field of novel high-performance low-noise permanent magnet motor design and manufacturing. Background Technology

[0002] Fractional-slot concentrated winding permanent magnet synchronous motors have been widely used in new energy vehicles, industrial servo systems, and aerospace drives due to their compact structure, high power density, high efficiency, and short-pitch winding characteristics. However, due to the special slot-pole matching and complex air gap magnetic flux density distribution, the electromagnetic excitation of this type of motor exhibits multi-directional coupling characteristics, which easily leads to significant electromagnetic vibration and noise problems.

[0003] Traditional research generally holds that motor vibration is mainly caused by radial electromagnetic force waves in the air gap. Therefore, related vibration suppression designs mostly focus on reducing radial magnetic flux density harmonics, such as by changing the slot pole fit, rotor skew poles, magnet shape, or adding auxiliary slots to suppress specific radial force harmonics. While these methods can weaken some dominant harmonics, they usually ignore the coupling relationship between tangential and radial electromagnetic forces.

[0004] With the increase in motor torque density and the increasing complexity of rotor structures, the influence of tangential electromagnetic force on stator vibration has become increasingly significant. Studies have shown that radial and tangential electromagnetic forces often have the same spatial harmonic order, and their phase relationship may lead to superposition or cancellation, thereby altering the overall vibration characteristics. Furthermore, according to the force translation theorem, the tangential electromagnetic force can be equivalent to a radial component acting on the stator tooth surface, which has an indirect driving effect on motor vibration. Therefore, strategies that simply suppress radial force have limitations.

[0005] Existing methods have also attempted to adjust the air gap magnetic field through rotor asymmetric structures, such as the rotor magnet biasing method proposed in the paper "Asymmetric Rotor Design of IPMSM for Vibration Reduction Under Certain Load Condition" (IEEE Trans. Energy Conversion, vol.35, no.2, pp.928–937, 2020), which can improve the radial harmonic components of a specific order. However, such methods require high manufacturing precision, are prone to introducing new low-order harmonics, and cannot achieve coordinated control of radial and tangential forces.

[0006] Therefore, existing technologies lack an effective design method that can systematically consider the coupling relationship between radial and tangential electromagnetic forces and achieve spatial redistribution of electromagnetic forces through structural modulation. In particular, in fractional-slot concentrated winding motors, how to use structural means to achieve synergistic suppression of multi-order electromagnetic harmonics remains a key challenge in current vibration and noise optimization design. Summary of the Invention

[0007] The purpose of this invention is to overcome the shortcomings of existing technologies that only suppress radial electromagnetic forces and ignore the coupling effect between tangential and radial electromagnetic forces. This invention proposes a vibration and noise suppression method for fractional-slot concentrated winding permanent magnet synchronous motors based on the redistribution of radial and tangential electromagnetic forces. This method is simple in structure and easy to manufacture. By introducing a chamfer structure at the stator tooth tip and combining it with the force translation theorem, the spatial distribution of air gap permeability is actively modulated, achieving a redistribution of the amplitude, phase, and space of radial and tangential electromagnetic forces. This allows the two forces to cancel each other out under the dominant harmonic order, significantly reducing the multi-order electromagnetic vibration and noise levels of the motor while maintaining essentially unchanged electromagnetic performance.

[0008] Specifically, the present invention is implemented using the following technical solutions:

[0009] A vibration and noise suppression method for fractional-slot concentrated winding permanent magnet synchronous motors based on radial-tangential electromagnetic force redistribution is described below:

[0010] Technical principle:

[0011] 1. Spatiotemporal distribution model of air gap magnetic flux density:

[0012] In the slotless state, the air gap radial magnetic flux density B r (θ,t) and tangential magnetic flux density B t (θ,t) can be expressed as:

[0013] ;

[0014] ;

[0015] Among them, B ar (θ,t) and B at (θ,t) represent the radial and tangential magnetic flux densities in the slotless state, respectively; λ(θ,t) is the relative permeability of the air gap. The relative permeability of the air gap satisfies:

[0016] ;

[0017] ;

[0018] ;

[0019] Where λ r0and λ s0 λ is the average permeability. rm and λ sn These represent the harmonic amplitudes of the rotor and stator permeability, respectively, β. m Z represents the rotor phase, Z represents the number of stator teeth, and m and n are positive integers.

[0020] 2. The uniformity of the spatiotemporal order of electromagnetic force:

[0021] Based on Maxwell's stress tensor, the air gap radial electromagnetic force f r (θ,t) and tangential electromagnetic force f t (θ,t):

[0022] ;

[0023] ;

[0024] Where μ0 is the permeability of air. The radial force and the tangential force have the same spatial and temporal order, only their phases differ. In the permanent magnet rotor motor that utilizes leakage flux, the dominant excitation is an electromagnetic force wave of the 10th order spatial frequency and 10 times the rotational frequency (10fr), while 25th and 35th order atypical harmonics also exist.

[0025] 3. Equivalent Action Model Based on Force Translation Theorem:

[0026] The tangential force is equivalent to the radial component:

[0027] ;

[0028] Synthetic radial excitation:

[0029] ;

[0030] Where F tri F represents the tangential equivalent radial force. ri Representing radial force, α is a geometry-dependent coefficient, A ti, φ ti This represents the amplitude and phase of the tangential force. When the phase difference is close to 180° under the 10th harmonic, the resultant force is significantly weakened.

[0031] Implementation steps:

[0032] Step 1: Determine the slot-pole configuration and structural form based on the motor's operating requirements. The target motor adopts a 12-slot, 10-pole fractional-slot concentrated winding structure with a star connection. The rotor uses a leakage flux utilization type structure, and the rotor permeability spatial period is 5.

[0033] Step 2: Establish a two-dimensional model of the motor, calculate the radial and tangential magnetic flux density distributions in the air gap under no-load conditions, and extract the main spatial and temporal harmonic components using a two-dimensional fast Fourier transform. Predict the electromagnetic force harmonic components that will have a significant impact on vibration and noise, and identify the dominant harmonic as the 10th order (10f). r ) and atypical harmonics of the 25th and 35th orders.

[0034] Step 3: Calculate the radial and tangential electromagnetic forces in the air gap. Use the force translation theorem to establish a radial-tangential electromagnetic force coupling model, and convert the tangential electromagnetic force into a radial component force acting on the stator tooth surface to form a comprehensive electromagnetic force distribution.

[0035] Step 4: Import the equivalent electromagnetic force loads under single-sided and double-sided chamfered structures into the structural finite element model, perform vibration simulation based on stator modal characteristics, and calculate the vibration acceleration spectrum based on harmonic response analysis. Select the frequency components with larger accelerations (10th, 25th, and 35th orders), analyze their electromagnetic force harmonic sources, and set them as optimization targets.

[0036] Step 5: Optimize all variable parameters of the single-sided and double-sided chamfers of the stator teeth (chamfer height 0-1mm, width 1-5mm). Obtain the optimal chamfer parameters through parameter scanning, which increases the amplitude of the tangential force under the 10th harmonic by about 30%, reduces the amplitude of the radial force by about 20%, and achieves a phase difference of 170°-180°, thus achieving the optimal motor vibration.

[0037] Furthermore, in step 2, the operating condition point selected in the actual vibration measurement of this motor is 1000 r / min under no-load conditions, and the target data analyzed is the electromagnetic force harmonic amplitude value at the no-load operating condition point.

[0038] Furthermore, in step 4, the vibration of the motor was measured, and the 10th, 25th, and 35th harmonic vibrations were very obvious in the spectrum. These three main electromagnetic force harmonic components were set as the final optimization targets.

[0039] Furthermore, in step 5, there are a total of 4 optimization variables in the single-sided chamfer and double-sided chamfer of the stator tooth tip. These 4 variables are mainly design variables related to the chamfer. Each chamfer has 2 variables, namely the chamfer width and the chamfer height.

[0040] Furthermore, in step 5, the parameter optimization method comprises the following steps:

[0041] 5-1) Select the initial design values ​​and variation range for the chamfer width (1-5 mm) and height (0-1 mm);

[0042] 5-2) The amplitude and phase of electromagnetic force under different chamfer parameters were calculated by finite element simulation, and the influence on the 10th harmonic was analyzed.

[0043] 5-3) Establish the relationship between the chamfering parameters and the response of the electromagnetic force target, and clarify the optimal parameter range;

[0044] 5-4) Select the optimal single-sided chamfer parameters: width 3.5 mm, height 0.67 mm, to achieve a 10th-order phase difference of 178°;

[0045] 5-5) Compare the optimal results with the original structure to analyze the vibration suppression effect and conduct experimental verification.

[0046] The present invention has the following beneficial effects:

[0047] (1) By introducing a chamfered structure to modulate the air gap magnetic field, the radial and tangential electromagnetic forces are redistributed using the force translation theory, which fundamentally reduces the electromagnetic excitation of the motor.

[0048] (2) The single-sided chamfer structure can form anti-phase cancellation under the dominant harmonics, which significantly weakens the synthetic vibration force on the stator tooth surface;

[0049] (3) This method has a simple structure, is easy to process, and has little impact on electromagnetic performance. It is suitable for fractional slot concentrated windings and other types of permanent magnet motors.

[0050] (4) Compared with the traditional method of only suppressing radial harmonics, the vibration acceleration of the present invention is reduced by 40% to 50%, which effectively reduces the overall vibration noise level of the motor.

[0051] In summary, the present invention provides a vibration suppression method for a fractional-slot concentrated winding permanent magnet synchronous motor based on radial-tangential electromagnetic force redistribution. This method modulates the air gap permeability by chamfering one side of the stator tooth tip, thereby achieving coordinated control of electromagnetic force amplitude and phase. It utilizes the force translation theorem to form anti-phase cancellation, significantly reducing vibration noise. The method is simple to process and has remarkable effects. Attached Figure Description

[0052] Figure 1 is a schematic diagram of the overall structure of the permanent magnet motor utilizing leakage flux of the present invention.

[0053] Figure 2 is a schematic diagram of the design of the stator single-sided chamfer and double-sided chamfer of the present invention.

[0054] Figure 3 is a comparison diagram of radial force harmonics and tangential force harmonics of the initial structure and the single-sided chamfer structure of the present invention. (a) is the radial force harmonic, and (b) is the tangential force harmonic.

[0055] Figure 4 is a diagram showing the coupling process of tangential and radial forces in the single-sided chamfered structure of the present invention.

[0056] Figure 5 shows the surface vibration acceleration diagrams of the initial structure and the single-sided chamfer structure of the present invention.

[0057] Figure 6 is a comparison diagram of the load torque of the initial structure and the single-sided chamfer structure of the present invention.

[0058] Figure 7 is a flowchart of the design of this invention. Detailed Implementation

[0059] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are for illustrative purposes only and are not intended to limit the invention. To more clearly illustrate the beneficial effects of this invention, an example of a fractional-slot concentrated winding spoke-wheel ferrite permanent magnet synchronous motor based on radial-tangential electromagnetic force redistribution will be used in the following description.

[0060] The overall structure of the motor described in this invention is shown in Figure 1. An outer casing (1) is provided, which is tightly fitted with the stator core (2) to fix the stator assembly and provide mechanical support. Armature windings (3) are embedded in the stator slots, using a concentrated winding configuration, distributed on both sides of the stator teeth. The rotor core (4) is located inside the stator core (2), forming a uniform air gap between them. Permanent magnets (5) are embedded inside the rotor core, employing a leakage flux utilization structure to ensure motor output performance. Both the stator and rotor cores are made of B50AH470 silicon steel sheets, and the permanent magnets are made of FB9H ferrite. The casing, shaft, and end caps are all made of aluminum alloy to reduce overall weight and improve heat dissipation. The chamfered structure (6) is machined at the stator tooth tip to adjust the air gap permeability distribution, achieving the redistribution of radial-tangential electromagnetic force and vibration suppression.

[0061] Step 1: Motor structure design and model establishment.

[0062] Based on the operational requirements of the target motor, the slot-pole configuration, winding connection method, and rotor structure are determined. The motor adopts a 12-slot, 10-pole structure, with a double-layer fractional-slot concentrated winding and a star connection. The rotor has a spoke-type structure, with each pole consisting of a pair of permanent magnets and a segmented south pole. Leakage flux is utilized using a circumferential magnetic bridge, and the rotor permeability spatial period is 5.

[0063] like Figure 1 As shown, the motor mainly consists of an outer stator 2, an inner rotor 4, permanent magnets 5, and stator windings 3. The stator 1 includes 12 stator slots and double-layer concentrated windings 3 distributed within them; the rotor 4 includes spokes and radially embedded ferrite permanent magnets. Motor parameters are shown in Table 1.

[0064] ;

[0065] Step 2: Air gap field modeling and chamfer structure design.

[0066] First, establish the air gap magnetic field according to the formula:

[0067] , ;

[0068] Among them B ar and B at λ(θ,t) represents the fundamental component of the air gap magnetic flux density in the slotless state; λ(θ,t) represents the relative air gap permeability.

[0069] Furthermore, the air gap permeability can be expanded into a Fourier form:

[0070] ;

[0071] ;

[0072] ;

[0073] Where λ r0 and λ s0 λ is the average permeability. rm and λ sn These represent the harmonic amplitudes of the rotor and stator permeability, respectively, β. m Z represents the rotor phase, Z represents the number of stator teeth, and m and n are positive integers.

[0074] like Figure 2 As shown, to achieve optimized modulation of the air gap magnetic field, a chamfer structure is introduced on the stator tooth tip surface. The chamfer includes two forms: single-sided chamfer and double-sided chamfer. The single-sided chamfer structure is machined on one side of the tooth tip along the rotor rotation direction; the double-sided chamfer structure is machined symmetrically on both sides of the tooth tip, with the angle and width remaining consistent. The chamfer structure alters the magnetic permeability distribution at the tooth tip, creating a local disturbance in the air gap permeability, thereby spatially modulating the air gap magnetic flux density waveform. The air gap permeability characteristics of double-sided and single-sided chamfers are determined using formulas.

[0075] The permeability of the double-sided chamfered air gap ratio satisfies:

[0076] ;

[0077] ;

[0078] ;

[0079] The permeability of the air gap with a single-sided chamfer satisfies:

[0080] ;

[0081] ;

[0082] ;

[0083] Where l is the chamfer width, k is the chamfer slope, and θ t Let θ be the tooth pitch angle. s For the opening of the groove.

[0084] Step 3: Radial-tangential electromagnetic force redistribution analysis.

[0085] The spatial-temporal distributions of radial and tangential magnetic flux density in the air gap were calculated using a finite element model, and the radial electromagnetic force F was obtained using the Maxwell stress tensor method. r and tangential electromagnetic force F t :

[0086] ;

[0087] ;

[0088] Where μ0 is the air permeability.

[0089] like Figure 3 As shown, a two-dimensional fast Fourier transform is used to decompose the force wave, and the harmonic orders of the radial and tangential electromagnetic forces are compared. The results show that the dominant electromagnetic force harmonics on the stator tooth surface of the motor are the 10th, 25th, and 35th orders. Simultaneously, a single-sided chamfer can increase the tangential force and reduce the radial force.

[0090] The tangential electromagnetic force component is converted into an equivalent radial component using the force translation theorem, thereby establishing a radial-tangential force coupling model:

[0091] ;

[0092] ;

[0093] Where F tri F represents the tangential equivalent radial force. ri Representing radial force, α is a geometry-dependent coefficient, A ti, φ ti This indicates the amplitude and phase of the tangential force.

[0094] The specific coupling process results are as follows: Figure 4 As shown, the tangential force and radial force are coupled into a unified radial force, and the influence of the tangential force on the vibration is analyzed. Furthermore, by adjusting the chamfer parameter, the two types of forces can be made to be close to 180° phase at the dominant harmonic, thus canceling each other out and reducing the combined electromagnetic excitation.

[0095] Step 4: Multiphysics simulation and vibration analysis.

[0096] The electromagnetic force distributions calculated for single-sided and double-sided chamfered structures are imported into the structural finite element model, and vibration response analysis is performed in conjunction with the stator core modal parameters.

[0097] Subsequently, the vibration acceleration of the motor's outer surface was calculated through harmonic response analysis, and the vibration acceleration diagram under no-load conditions was obtained. The results are as follows: Figure 5 As shown, in the single-sided chamfered structure, the 10th-order vibration acceleration decreases from 1.28 m / s² to 0.72 m / s². 2 The amplitudes of the 25th and 35th harmonics decreased by approximately 30% and 27%, respectively, and electromagnetic vibration decreased by approximately 45%.

[0098] Step 5: Parameter optimization and performance verification.

[0099] Based on the simulation results, parameter scanning of the chamfer angle and width was performed, yielding optimal chamfer parameters of 3.5mm width and 0.67mm height. Under these conditions, the phase difference between the radial and tangential electromagnetic forces is approximately 170°, resulting in near-anti-cancellation. Compared to a motor without chamfers, the motor employing the structure of this invention exhibits significantly reduced vibration amplitude under the same operating conditions. Furthermore, because the chamfer structure has minimal impact on the main magnetic circuit, ... Figure 6 As shown, the average torque of the motor decreased by no more than 1%, verifying the effectiveness of the method in balancing electromagnetic performance and low noise characteristics. Figure 7 This is a flowchart illustrating the overall design process of the present invention, showing the interrelationships between the various steps.

[0100] As can be seen from the above embodiments, this invention achieves spatial redistribution of electromagnetic forces by setting a chamfer structure at the stator tooth tip and combining it with the force translation theorem to coordinate and control the radial and tangential electromagnetic forces, thereby forming a vibration cancellation effect under the dominant harmonic order. Compared with the traditional method of only suppressing radial forces, this invention has a simple structure, low cost, and easy processing, and can significantly reduce electromagnetic vibration and noise levels while maintaining high torque output performance.

[0101] In this specification, the terms "one embodiment," "some embodiments," "exemplary embodiments," "specific embodiments," or "several embodiments," etc., indicate that the technical features, structures, or parameters described in connection with that embodiment can be applied to at least one implementation of the present invention. Identical or similar features between different embodiments may be combined with each other unless they contradict each other.

[0102] Although this specification has described the invention in detail with reference to the accompanying drawings and specific embodiments, those skilled in the art should understand that various modifications, equivalent substitutions or variations can be made to the embodiments without departing from the spirit and essence of the invention, and all such modifications, equivalent substitutions or variations should fall within the scope of protection claimed by the invention.

Claims

1. A vibration noise suppression method for a fractional-slot concentrated-winding permanent-magnet synchronous motor, characterized by, The method comprises the following steps: Step 1, according to the target performance of the motor and the operation condition requirements, determine the slot-pole matching and winding structure; Step 2, establish electromagnetic size equations according to the motor power, speed and the assembly requirements of the stator and rotor, determine the main parameters of the stator outer diameter, rotor outer diameter, shaft diameter, core length, air gap length, pole arc coefficient and magnetic steel size; establish a two-dimensional electromagnetic model by using finite element software, and perform electromagnetic characteristic calculation under no-load and rated conditions respectively to verify that the electromagnetic performance meets the design index; Step 3, establish a three-dimensional structure model of the motor, consider all assembly components such as the shell, stator, winding, rotor, shaft, end cover and bearing, and set mutual contact constraints; perform modal analysis on the complete motor model to obtain the first ten orders of natural frequency and modal shape, and provide boundary conditions for subsequent vibration characteristic analysis; Step 4. Calculate the radial flux B in the air gap based on the finite element model r (θ, t) and the tangential flux B t (θ, t) distribution; Step 5, based on the force translation theorem, convert the tangential electromagnetic force component into an equivalent radial force, which is expressed as: ； An equivalent action model of the tangential force on the radial vibration of the stator surface is established, and a vector superposition model is established to realize the unified analysis of the radial and tangential electromagnetic forces, and the vector superposition model can be expressed as: ； where F tri represents the tangential equivalent radial force, F ri represents the radial force, a is a geometry dependent coefficient, A ti, φ ti represents the amplitude and phase of the tangential force; Step 6, set a chamfer structure on the stator tooth top to modulate the air gap permeability distribution, and the wave form of the conventional air gap permeability without chamfer is expanded in the Fourier form: ； ； ； where λ r and λ s represent the permeability of the rotor and stator, respectively, λ r0 and λ s0 are the average permeability, λ rm and λ sn are the rotor and stator permeability harmonic amplitudes, respectively, β m is the rotor phase, Z is the number of stator teeth, ω0is the mechanical angular velocity, θ represents the mechanical angle, t represents time, and m, n are positive integers; The chamfer structure includes single-sided chamfer and double-sided chamfer, and the air gap flux waveform can be changed by adjusting the chamfer parameters; Step 7, calculate the electromagnetic force harmonic components under the double-sided chamfer and the single-sided chamfer respectively by using the finite element method, and compare and analyze the radial and tangential electromagnetic force amplitudes and phase differences under the main spatial harmonics of the two structures; Step 8, establish a vector superposition model of the radial and tangential electromagnetic forces, and realize electromagnetic force redistribution by adjusting the phase difference, so that the tangential electromagnetic force is opposite to the radial electromagnetic force at the dominant harmonic frequency, thereby weakening the combined radial vibration force; Step 9, apply each order electromagnetic force harmonic load to the stator tooth surface of the three-dimensional structure model, combine the modal analysis results to perform vibration response simulation, and calculate the motor surface vibration acceleration and sound power spectrum by using the acoustic boundary element method; Step 10, compare and analyze the vibration acceleration amplitudes under the single-sided chamfer and the double-sided chamfer structure, determine the optimal chamfer structure scheme, and the optimal chamfer structure shows significant vibration amplitude reduction at the main order and non-typical order, thereby realizing significant electromagnetic vibration and noise suppression in the full-speed range of the motor.

2. The method of claim 1, wherein, The chamfer in step 6 is arranged along the inclined edge in the rotation direction, and by adjusting the chamfer scheme and the chamfer parameters, the tangential electromagnetic force can be increased, the radial electromagnetic force can be reduced, and the phase difference between the two can be close to the opposite phase, thereby forming a vibration cancellation effect.

3. The method of claim 1, wherein, The vibration simulation in step 9 adopts the finite element and boundary element coupled calculation method, loads the radial and tangential electromagnetic force harmonics respectively, obtains the vibration acceleration frequency spectrum of the motor under the rated no-load speed, and the optimal chamfer structure makes the dominant order vibration acceleration significantly reduced and the vibration amplitude greatly reduced.

4. The method of claim 1, wherein, The analysis and optimization process can be applied to permanent magnet synchronous motors with asymmetric rotor structures, including leakage utilization type, radiation type embedded type and V type embedded type rotor structures, and through radial-tangential electromagnetic force redistribution, a wide spectrum of vibration and noise suppression is achieved.

5. The method of claim 1, wherein, In step 4, the air gap magnetic flux density distribution satisfies the following formula: ； ； where B ar and B at is the fundamental component of air-gap flux density in the slotless state; λ(θ, t) is the relative air-gap permeability; the main spatial and temporal harmonic components are extracted by two-dimensional fast Fourier transform; the radial electromagnetic force f r (θ, t) and the tangential electromagnetic force f t (θ, t) are calculated by using the Maxwell stress tensor: ； ； Where μ0 is the air permeability.

6. The method of claim 1, wherein, In step 6, the air gap ratio permeance of the double-side chamfer can be expressed as follows: ； ； ； The air gap ratio permeance of the single-side chamfer can be expressed as follows: ； ； ； where λ s (θ) is the relative permeance, λ s0 (θ) is the average relative permeance, λ sν and λ su represent the amplitudes of the cosine and sine components of the relative permeability harmonics, respectively, Z is the number of stator teeth, θ represents the mechanical angle, v and n are the permeance order, l is the chamfer width, k is the chamfer slope, θ t is the tooth pitch angle, and θ s is the slot opening.

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