An Improved Model for Five-Phase Induction Motor Based on Magnetic Noise Reduction Part II: Pole-Slot Scheme

Abstract

For the reduction in the electromagnetic noise level of three-phase induction motors, many empirical rules and analytical models have been established to select the matching scheme of pole and slot, but they are not fully applicable to five-phase squirrel cage induction motors (FSCIM). In this paper, combined with the slot-number phase diagram (SNPD), and the electromagnetic force-vibration-acoustic analytical model deduced in Part I, the influence of pole-slot schemes, including five-phase regular-size phase-belt and fractional-slot winding, on magnetic noise is analyzed. The feasibility of electromagnetic noise prediction is verified by finite element simulation and experiments. Taking a 4 kW FSCIM as a prototype, noise prediction is carried out for all the slot-number matching schemes with pole pairs not exceeding three. For two noise reduction targets, which reduce the maximum single-frequency noise in the steady-state operation and the average noise during startup, the pole-slot numbers matching rule of FSCIM is given. This improved model is also applicable in different power ranges for the noise reduction design of five-phase motors.

1. Introduction

With the promotion and popularization of new energy vehicles, in addition to power density and fault tolerance performance, the electromagnetic vibration noise of electric motors, which is closely related to the noise, vibration, and harshness (NVH) characteristics of electric vehicles, has also received more and more attention. Similar to the three-phase motor, the pole-slot coordination is an important factor affecting the electromagnetic vibration characteristics of the motor. Some of the original rules are in [1], and numerous empirical rules have been developed in the ensuing decades. An exhaustive list of these laws can be found in Timar’s book [2]. However, these results mainly focused on three-phase motors, without the consideration of the natural frequency and modal characteristics of the motor structure. Additionally, some of them are based on electromagnetic torque pulsation as the limiting condition, which may not be suitable for reducing audible magnetic noise [3]. Due to the difference in the number of phases, the air-gap magnetic density and electromagnetic force harmonic of the five-phase squirrel cage induction motors (FSCIM) are various.

To design an FSCIM for driving, it is necessary to enhance NVH characteristics. In almost all induction machines, noise originates from aerodynamic, mechanical, and electromagnetic problems [4,5,6]. Aerodynamic and mechanical problems, which mainly result from turbines and assembly errors, can be ignored in the design stage. However, electromagnetic vibration levels can directly affect the NVH characteristics and cause failures, such as bearing failure and insulation breakdown [7]. Magnetic pull force [8], torque ripple [9], cogging torque [9], and unbalanced magnetic force [8] are the main electromagnetic sources of NVH characteristics. Improvement of the overall force characteristics is required for improving the NVH characteristics. Additionally, optimizing the individual force characteristics while improving the overall electromagnetic force characteristics is a better choice to improve the NVH characteristics. The electromagnetic vibration characteristics of the five-phase motors are similar to those of the three-phase motors. The pole-slot number scheme has a decisive effect on the magnetic noise [10,11,12,13]. Obviously, it is not suitable to directly apply the pole-slot matching of the three-phase motor to FSCIM.

A 40/30 slot, four poles, and an FSCIM are used to study the influence of saturation on the air-gap flux density waveform [14]. Wang Dong et al. selected three five-phase motors with 60/38 slots to form a fifteen-phase induction motor, and analyzed the air-gap magnetic potential of the induction motor under non-sinusoidal power supply conditions [15]. A comparative analysis of the operating characteristics between the three-phase and five-phase induction motors under the same structural size, that is, 30/44 slots and 2-poles induction motor, is discussed in [16]. Pereira LA et al. analyzed the mathematical model of the five-phase induction motor, deduced the self-inductance and mutual inductance of the stator and rotor, and calculated the time and space harmonics of the air-gap flux density [17,18]. The effect of applying stator shifting to five-phase winding to suppress the effect of the slot harmonics by doubling the number of slots is investigated [19]. Based on the measured sample data, a new radial vibration model is proposed, consisting of a vibration acceleration impulse model and a natural oscillation model [20]. The vibration and noise levels in a permanent magnet synchronous motor with different slot-pole combinations are discussed, mainly through the FEM to establish the analytical model [21]. An analytical model of the acoustic behavior of pulse-width modulation (PWM) controlled induction machines is applied to a three-phase fractional-slot winding machine. However, the acoustic radiation is simplified as a 2D cylindrical shell model [22,23]. Reference [24] established the electromagnetic vibration and noise model of the three-phase induction motor based on the acoustic model of the infinitely long ring, and carried out the optimization study of the slot-number scheme, but did not consider the influence of the pole-pair-number and the axial modes of the stator frame in an acoustic radiator. Reference [25] aimed at reducing the resonance noise of an evaporative cooling motor induced by an electromagnetic and two-phase flow based on the fluid-structure coupling theory. The subdomain method is used to optimize the noise, vibration, and harshness (NVH) characteristics of a permanent magnet synchronous motor. The predecessors mainly analyzed the operating characteristics and control strategies of the existing five-phase motor or those created by re-embedding the stator winding of the three-phase motor. However, there rarely are selection basis and parameter optimization processes of pole-slot number schemes for five-phase induction motors.

This paper aims at the optimization design of the electromagnetic noise of an FSCIM, as shown in Figure 1. Firstly, the slot-number assignment of the five-phase regular-size phase-belt winding is detailed via the slot-number phase diagram (SNPD) [26,27]. Then, the expression of stator magnetomotive force (MMF) can be given by the superposition of a single conductor or a coil. Next, an improved analytical model of the induction machine’s electromagnetic force-vibration-acoustic radiation is proposed. Based on these mathematical models, the influence of time and space harmonics of electromagnetic forces on vibro-noise will be considered as a whole. The accuracy of this model for predicting magnetic noise phenomena is validated by a prototype at different stages (natural frequency, vibration, and sound power level) by numerical methods and tests. Finally, focused on two different low electromagnetic noise targets, the magnetic noise level of every pole-slot number scheme of five-phase induction motors with less than four pole pairs is simulated. Taking a 4 kW FSCIM with an outer diameter of 175 mm as a prototype allows for recommending schemes for low-noise slot matching.

2. Electromagnetic Noise Analytical Prediction Model

2.1. Subordination of Slot-Numbers and Phase

SNPD is a useful method for expressing the winding MMF that quickly obtains the multi-phase symmetrical winding distribution and is convenient for nested programming. The tabular form (including Z₁ positive slots and Z₁ negative slots) presents the unit space vector distribution of the MMF generated in each slot. For the five-phase winding, the normal connection winding can be divided into 36° phase-belt winding, regular-size phase band winding, and 72° phase-belt winding. As shown in Figure 2, the positive phase-belt is expanded by L slots, as the negative phase-belt reduces L slots. When L = 0, the common 36° phase-belt winding is given.

Assuming that the slot-number J is located at the R-th cell of the phase diagram when the phase of the slot-number is represented by R, the following formula should be satisfied:

(1)$R^{*}(J)=D(|J|-1)+1+m_{1}N\frac{1-\mathrm{sgn}(J)}{2}$

(2)$R(J)=\mathrm{mod}(R^{*}(J),2m_{1}N)$

where $q=\frac{N}{D}$, N, and D are co-prime, mod( ) indicates the remainder, sgn( ) is the signum function, and sign of J only indicates the current flow in the slot.

In terms of Figure 2, the phase of the slot can be determined according to the phase R of the slot-number. If the number of cells occupied by the positive and negative phase-belt are N_p + L and N_p − L, respectively, the slots-number that belong to the β-th phase (1–5 corresponding to the A–E phase, respectively) J, can be assigned according to the following rules:

(3)$2(\beta -1)N+L\frac{1-\mathrm{sgn}(J)}{2}\le R(\beta ,J)\le (2\beta -1)N+L\frac{1+\mathrm{sgn}(J)}{2}$

According to the aforementioned, the expression of the winding coefficient can be deduced for each pole-slot match. Taking the two motors shown in Table 1 as an example, the phase-belt distribution can be obtained as shown in Figure 2 by (3).

It can be seen from Figure 2 (a) that for the M2 motor, the subordinations of the slot-numbers of L = 0 and L = 1 are consistent. So far, by combining (1), (2), and (3), the symmetrical distribution of the five-phase normal windings and the affiliation of each slot-number can be determined only via Z₁ and p. If the center line of the A-phase winding is taken as the origin, the winding function of the five-phase normal winding can be obtained from the slot-number distribution R(m) and (4) by the superposition principle.

(4)$W{F}_s({\theta }_s)=\left\{\begin{array}{l}{\displaystyle \sum _{v=1}^{\infty }\frac{k_{sv}}{\pi v}\sin v{\theta }_s,\mathrm{single}\mathrm{conductor}}\\ {\displaystyle \sum _{v=1}^{\infty }\frac{2N_c}{\pi v}{k}_{sv}{k}_{yv}\cos v{\theta }_s,\mathrm{single}\mathrm{coil}}\end{array}\right.$

where N_c is the number of turns of the coil; k_sv and k_yv are the slotting coefficient and short distance coefficient of v pair poles harmonics, respectively.

2.2. Analytical Calculation of Electromagnetic Force

For induction motors, the MMF generated by the rotor is often much smaller than the stator in the case of no load, not resulting in new electromagnetic resonance phenomena [28]. Therefore, the radial component B_r of the magnetic flux density can be expressed as [29,30]:

(5)$B_r(t,{\theta }_s)=f_{mmf}^s(t,{\theta }_s)\Lambda (t,{\theta }_s)$

(6)$\Lambda (t,{\theta }_s)={\Lambda }_0+{\displaystyle \sum _{k_1}{\lambda }_{k_1}}+{\displaystyle \sum _{k_2}{\lambda }_{k_2}}+{\displaystyle \sum _{k_1}{\displaystyle \sum _{k_2}{\lambda }_{k_1}{\lambda }_{k_2}}}$

The specific expressions of each part of Λ have been given in detail in [29,31]. Noticeably, the effects of both sides slotting in Λ are proportional to the width of the slot opening. Both ${\lambda }_{k_1}$ and ${\lambda }_{k_2}$ are inversely proportional to the harmonic order of the permeance k₁ and k₂. The MMF of the stator is obtained when the sequential five-phase currents pass through the windings. Substituting (3), (4) and (5) into Maxwell’s radial force expression (7), the radial electromagnetic forces generated by all radial magnetic flux harmonics can be determined.

(7)$P_{eR}(t,{\theta }_s)=B_r^2(t,{\theta }_s)/2{\mu }_0$

Considering only the major force waves in magnetic noise, that is, the low-order force waves P_belt sourced from the interaction between the harmonics of phase-belt of stator MMF and the tooth harmonics of air-gap permeability. The main force characteristics are given in Table 2.

It can be seen from Table 2 that the pole-slot matching scheme has a decisive influence on the force wave characteristics. The number of stator and rotor slots together affects the force wave order, while the frequency is mainly affected by the Z₂, p, s, and supply frequency f₁. Furthermore, five-phase motors are mostly powered by PWM with rich harmonics, so the frequency range of Maxwell forces is likely to cover the natural frequency of lower modes of the stator system.

2.3. Electromagnetic Vibration-Acoustic Radiation Model

The stator system mainly consists of an iron core, windings–tooth, and housing, which can be regarded as equivalent rings, respectively. The equivalent model can be divided into finitely and infinitely cylindrical shells for analytical calculation of vibration and noise based on Table 3.

For an infinite cylindrical shell, the analytical expressions for vibration and acoustics are given in [32,33]. In another case, a more detailed description of the finite shell can be found in the submitted paper Part I.

Due to the large stiffness of the motor shaft, the natural frequencies of the same circumferential mode are relatively close, and the influence of the axial mode on the natural frequency is smaller than that of the circumferential mode. Therefore, the sound power of different axial modes under the same circumferential mode excited by the m-order force wave can be considered uniformly. The noise of the low axial orders of the same circumferential mode is summed up by using the principle of modal superposition:

(8)$L_W(n,m)=10 {\log }_{10}({\displaystyle \sum _{a=1}^3{W}_m(n,a,m)/W_0})$

(9)$L_{WA}(n,m)=10 {\log }_{10}({\displaystyle \sum _n{10}^{0.1\left(L_w\left(n,m\right)+\Delta L_A\left(n\right)\right)}})$

where ΔL_A(n) is sound frequency dependent A-weighting factor.

2.4. Model Validation

The model has been validated at different stages (natural frequency, vibration, and sound power level) by numerical methods and tests in the submitted paper Part I and [22,28,34].

Furthermore, in this paper, the experimental measurement and analysis of the vibration-noise of the prototype in the transient process are carried out, and the results further verify the accuracy of the analytical model. The layout of the vibration acceleration sensors (#A1–#A5) and sound sensors (#S1–#S3) is shown in Figure 3, and an overall layout of the test platform is given in Figure 4.

According to the principle of frequency conversion speed regulation, the starting process is divided into 10 equal processes for vibration and noise measurements. The five-phase currents and speed curves over time are shown in Figure 5.

Theoretically, the stable speed of 10 stages should be proportional to the power supply frequency, but due to the idling loss, the speed is not strictly proportional to the frequency. The stator current shows a trend of increasing-decreasing-increasing. Meanwhile, the vibration and noise results obtained by sensors placed as in Figure 3 are shown in Figure 6, Figure 7, Figure 8 and Figure 9.

As shown in Figure 6, acceleration amplitudes and trends between the top and bottom are basically consistent. However, the side vibration amplitude is almost half of the top, and the change trend with the supply frequency is different, but consistent with the trend of the current.

The natural frequency of the stator system is given in the submitted paper Part I, so this paper will use it as known data. Through spectrum analysis of the acceleration at each stage in Figure 7, it is found that the main vibration frequency is concentrated at approximately 5000 Hz and 1000 Hz, which corresponds to the modes (3, 1), (3, 2), (1, 4), (2, 4) and (3, 4) of the stator system. For vibration acceleration, it is mainly reflected in the circumferential modem, that is, m = 1, 2, and 4. In terms of Figure 7j, the analytical model is used to calculate the vibration acceleration of the test motor in main modes. From Table 4, it is obvious that mode (3, 2) contributes the most to the vibration, while mode (3, 1) is the opposite, even though the natural frequencies of both are very close. Furthermore, all three modes with m = 4 have contributions to the vibration, hence the main modes are confirmed to m = 2 and 4 for the test motor, which is consistent with Figure 10.

From Figure 8, it is found that the sound pressure intensities and trends in the top, side, and axial directions are more consistent than they are in vibration. Focused on side sensor #S2, 10 stages of spectrum analysis are performed in Figure 9.

With the increase in the power supply frequency and rotation speed, the noise in the low-frequency region (500–2000 Hz) gradually increases and fluctuates. This frequency band that does not appear in the frequency spectrum of vibration acceleration is related to the supply frequency and speed. The mechanical noise corresponding to speed is mainly generated by the bearing connection, and the part related to supply frequency is the result of the resonance of the PWM module and the heat sink. Additionally, the magnetic noise of the test motor also fluctuates with the operation condition, and the amplitude spectrum also agrees with the vibration in both the analytical model and experiment.

3. Low-Noise Pole-Slot Optimal Combination

3.1. Low-Noise Targets

At present, the five-phase motor mainly adopts PWM variable frequency speed control. During the process from start-up to stable operation, the power supply frequency rises linearly from 0 to f₁, during which a rich force wave will be generated. In order to avoid the resonance between the low-order force wave P_belt shown in Table 2 and the circumferential mode of the stator, the following conditions must be fulfilled:

(10)$f_1[k_2{Z}_2(1-s)/p+l]<f_m,m=|k_2{Z}_2-k_1{Z}_1+lp|$

According to the noise reduction requirements in different environments, two noise optimization goals are set, respectively:

• (1). The minimum total sound power level of each frequency in rated condition:

(11)$\min ({\displaystyle \sum _m{L}_{WA}(f_m)})$

• (2). The sound power amplitude of the maximum noise generated during the start-up is the smallest:

(12) $\min (\max (L_{WA}(f_m)))$

For the pole-slot matching scheme, the above two optimization objectives should be considered, and it should be determined according to the application environment of the motor.

3.2. Pole-Slot Matching Strategy

When the power demand of a certain type of motor is determined, its outer diameter is also determined according to the national standard. On the basis of the aforementioned analytical model, all possible slot matching schemes are traversed, and the noise radiation in the process of frequency conversion and speed regulation is analyzed and calculated. The low-noise slot fit solution can be screened out. This paper takes the test motor shown in Table 3 as the object, and comprehensively considers the outer diameter of the motor, the principle of fewer slots and near-slots, and the symmetry conditions of multi-phase motor windings [10]. All cases of symmetrical distribution of stator windings in the range of Z₁ ∈ [5, 50], Z₂ ∈ [2, Z₁], and p ∈ [1, 3], and the following assumptions are made:

• (1). The slip ratio changes linearly.

• (2). In each case, five-phase sequential currents of the same amplitude are passed through the stator windings.

• (3). The magnitude of the air-gap MMF generated by the stator winding remains unchanged, so the N₁k_w1 under various schemes remains approximately unchanged. It should be noted that the weight of the winding and the stiffness will change.

(13)$N_1{k}_{w1}=const$

• (4). In order to control the variables, the influence of slot opening and slot type is not considered when changing the number of stator and rotor slots. It can be known from the amplitude of air-gap permeability [29,31]. It is necessary to keep Z₁b₀₁ and Z₂b₀₂ unchanged.

(14) $Z_1{b}_{01}=const,Z_2{b}_{02}=const$

• (5). Excluding the influence of magnetic saturation, the air-gap permeance does not contain the harmonic content caused by saturation.

• (6). The slot number Z₁ is greater than Z₂.

The electromagnetic noise results of the slot fitting scheme are obtained by calculation, which include the maximum and average noise levels radiated by the motor during operation under each scheme, and the noise radiation under each modal resonance. The predicted results of the test motor are shown in Figure 10.

As can be seen from Figure 10, the sound power level in (b) during the starting process is higher than that in (a) under steady-state. This is because the frequency of force waves is much more abundant during the starting process, which is more likely to result in stator system resonance. Firstly, consistent with the previous experiment’s results, the magnetic noise is mainly generated from modes m = 2 and 4 on the test motor with 30/26 slots. In addition, both the average magnetic noise level in the starting process and the maximum noise level in the steady-state show discrete changes with the increase in the number of rotor slots. As expected, each stator mode has different effects on the overall noise level. For the five-phase motor with Z₁ = 30 and p = 1, the resonance effect at m = 2 (that is, the elliptical mode) is the strongest and dominant in both operating conditions. It is worth noting that for steady-state operation, the effects of modes m = 2 and 4 tend to have the same trend, which is also consistent with the previous conclusions of the test motor, but the effects of the two modes are quite the opposite during start-up. In summary, the initial design should focus on avoiding the slot matches that produce such modal resonance.

3.3. Optimal Pole-Slot Scheme

By analyzing the noise prediction results of all schemes, for the two different optimization objectives in Section 3.1, the quietest combinations between Z₂ and p in each five-phase symmetrical stator winding are screened out, as shown in Table 5, Table 6 and Table 7. According to the noise level, the rotor slots number are arranged in increasing order from left to right. From the above three tables, it can be found that no matter what the parity of Z₁ and Z₂ are, there will always be some low-noise combinations. For inter-slot motors (underlined in the table), the quiet slot combinations will be significantly different from the fractional-slot situation due to the change in the flux density harmonics content. Generally speaking, the quietest rotor slot number of a fractional-slot winding will be less than that of the integer-slot case. If Z₁ and Z₂ are both a multiple of 2p, the expression of force wave order in Table 2 can be transformed into (15), hence the main force orders should be a multiple of 2p which is in agreeance with Figure 10.

(15)$m=2p\left(k_2{q}_2-k_1{q}_1+0.5l\right)$

It should be noted that the several limitations of the pole-slot combination mentioned above are: it only considers the symmetrical winding with a number of pole pairs less than four, and it calculates the magnetic noise level without consideration of the effects of saturation, slot shape, and PWM harmonics. However, there is no doubt that this prediction model can be applied to five-phase induction motors with other power, size, and power supply conditions, in the initial design process to avoid strong magnetic noise and vibrations due to slotting harmonics.

4. Conclusions

Aiming at the relationship of the FSCIM between the electromagnetic noise characteristics and the pole-slot numbers match, this paper deduces and improves the radial electromagnetic force-vibration-noise radiation model based on the analysis of the five-phase symmetrical windings and the acoustic model in finite cylindrical shells. This model can comprehensively consider the natural frequency of the system composed of the three parts of the stator superimposed, as well as the acoustic radiation characteristics of the circumferential and axial modes of the finite cylindrical shell. Therefore, the results are more in line with the real situation of the electromagnetic noise of the motor. The accuracy of the improved model was verified by simulation and experiments at multiple stages under the conditions of a given power, supply frequency, and slot type. The noise prediction of as many slots as possible was carried out, and some quiet pole-slots for the FSCIM test were selected by different noise optimization objectives. The results reflect the effects and rules of pole-slot matching for noise reduction in five-phase induction motors.

Although the matching results in this paper have some limitations, which are that the results only reflect the effect of pole-slot numbers without the consideration of the slot geometries and PWM harmonics. However, they can still be applied to motors with similar natural frequencies, and the analytical noise prediction model can be applied to FSCIM with other requirements. To avoid severe resonance and noise caused by electromagnetic force waves, the model is still beneficial for the selection of pole-slot numbers during motor initial design.

Future works should focus on the multi-objective optimization algorithms for low-noise optimization design, comprehensively considering the pole-slot scheme, and slot geometries.

Author Contributions

Writing—original draft preparation, H.C.; validation, Y.X.; software, H.X.; supervision, J.Z.; data curation, S.Y. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

Nomenclature

Electrical notations:
Br	radial air-gap magnetic density
θs	angular position on the circumference
μ0	air magnetic permeability
fn	frequency of electromagnetic force
m	force wave order, circumferential mode
f1	fundamental supply frequency
$f_{mmf}^s$	stator magnetomotive force
WFs	stator winding function
Pbelt	major electromagnetic force
Mechanical and acoustic notations:
Z1	stator slots number
Z2	rotor slots number
D1	stator core outer diameter
Di1	stator core inner diameter
p	pole pair number
v	harmonic pole pairs
ksv	slotting coefficient
kyv	short distance coefficient
s	slip
δ	air-gap length
m1	phase number, m1 = 5
q	number of slots per pole per phase
b01	stator slots opening width
b02	rotor slots opening width
Nc	number of turns
a	axial mode
l	length of the housing
R	radius of the housing
Wm	sound power
W0	reference sound power
LW	sound power level

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Figures and Tables

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Figure 1. Model of five-phase cage induction motor.

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Figure 2. SNPD of five−phase fractional slot size phase−belt winding. (a) M1 motor; (b) M2 motor.

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Figure 3. Schematic diagram of sensors layout.

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Figure 4. FSCIM drive control and noise measurement platform. (a) Five-phase PWM drive system; (b) vibration-noise measurement system.

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Figure 5. Five−phase current and speed curves from start−up to steady-state.

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Figure 6. Vibration acceleration of three measuring points on the frame.

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Figure 7. The vibration acceleration spectrum of the top point #A1 in 10 stages. (a) Stage 1, f = 0.1 f1; (b) Stage 2, f = 0.2 f1; (c) Stage 3, f = 0.3 f1; (d) Stage 4, f = 0.4 f1; (e) Stage 5, f = 0.5 f1; (f) Stage 6, f = 0.6 f1; (g) Stage 7, f = 0.7 f1; (h) Stage 8, f = 0.8 f1; (i) Stage 9, f = 0.9 f1; (j) Stage 10, f = f1.

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Figure 7. The vibration acceleration spectrum of the top point #A1 in 10 stages. (a) Stage 1, f = 0.1 f1; (b) Stage 2, f = 0.2 f1; (c) Stage 3, f = 0.3 f1; (d) Stage 4, f = 0.4 f1; (e) Stage 5, f = 0.5 f1; (f) Stage 6, f = 0.6 f1; (g) Stage 7, f = 0.7 f1; (h) Stage 8, f = 0.8 f1; (i) Stage 9, f = 0.9 f1; (j) Stage 10, f = f1.

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Figure 8. Vibration acceleration of three measuring points on the frame.

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Figure 9. The sound pressure spectrum of the top point #S2 in 10 stages. (a) Stage 1, f = 0.1 f1; (b) Stage 2, f = 0.2 f1; (c) Stage 3, f = 0.3 f1; (d) Stage 4, f = 0.4 f1; (e) Stage 5, f = 0.5 f1; (f) Stage 6, f = 0.6 f1; (g) Stage 7, f = 0.7 f1; (h) Stage 8, f = 0.8 f1; (i) Stage 9, f = 0.9 f1; (j) Stage 10, f = f1.

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Figure 9. The sound pressure spectrum of the top point #S2 in 10 stages. (a) Stage 1, f = 0.1 f1; (b) Stage 2, f = 0.2 f1; (c) Stage 3, f = 0.3 f1; (d) Stage 4, f = 0.4 f1; (e) Stage 5, f = 0.5 f1; (f) Stage 6, f = 0.6 f1; (g) Stage 7, f = 0.7 f1; (h) Stage 8, f = 0.8 f1; (i) Stage 9, f = 0.9 f1; (j) Stage 10, f = f1.

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Figure 10. A Z1 = 30 FSCIM electromagnetic sound power level in function Z2 for both operation conditions. (a) Rated condition; (b) start-up condition.

Table 1

Generic five-phase fractional slot motors M1 and M2.

Symbol	M1	M2
Number of stator slots Z1	30	40
Number of rotor slots Z2	26	32
Number of pole pairs p	2	3
Number of slots moved L	1	2

Table 2

Characteristics of phase belt force wave P_belt.

Symbol	Frequency fP	Force Wave Mode m	Remark
Pbelt	fn[k2Z2(1- − s)/p + l]	k2Z2- − 2m1pk1 + lp	l = 0, ±2

Table 3

Criteria for type of analytical model.

Condition	Type of Equivalent Shells
ml ≥ 10 aR	Infinitely cylindrical shell
ml < 10 aR	Finitely cylindrical shell

Table 4

Frame surface vibration acceleration a_a,m (unit is m/s²) by improved analytical method.

aa,m (m/s2)	Mode a
Mode m	1	2	3
0	1.88	1.11	0.41
1	0	0	0
2	3.65	3.49	13.96
3	0	0	0
4	4.69	2.10	3.57

Table 5

Low-noise pole-slot numbers scheme for p = 1.

Z 1	Z2 of Target (10)	Z2 of Target (11)
5	4, 2, 3	3, 2, 4
10	5, 9, 3, 6, 4	7, 3, 8, 4, 2
15	5, 14, 6, 9, 10	5, 10, 4, 12, 13
20	16, 13, 2, 17, 9	13, 19, 18, 9, 3
25	24, 5, 21, 12, 19	23, 7, 18, 14, 9
30	21, 13, 17, 26, 27	19, 23, 21, 18, 27
35	34, 32, 5, 26, 4	34, 28, 22, 25, 5
40	32, 38, 24, 2, 37	31, 38, 23, 36, 39

Table 6

Low-noise pole-slot numbers scheme for p = 2.

Z 1	Z2 of Target (10)	Z2 of Target (11)
5	2, 3, 4	2, 3, 4
10	2, 5, 7, 3, 8	8, 2, 4, 6, 7
15	2, 8, 13, 4, 6	10, 9, 2, 4, 11
20	12, 4, 18, 10, 8	12, 4, 16, 8, 10
25	23, 15, 20, 18, 10	24, 20, 23, 10, 8
30	20, 10, 24, 23, 15	10, 20, 23, 29, 18
35	23, 24, 30, 32, 7	23, 30, 16, 32, 10
40	36, 32, 30, 23, 12	32, 36, 12, 4, 16

Table 7

Low-noise pole-slot numbers scheme for p = 3.

Z 1	Z2 of Target (10)	Z2 of Target (11)
5	2, 4, 3	3, 2, 4
10	6, 2, 8, 4, 3	2, 5, 6, 4, 8
15	12, 4, 2, 14, 8	5, 3, 10, 2, 11
20	19, 10, 18, 6, 12	10, 12, 13, 5, 11
25	23, 19, 18, 24, 6	13, 3, 6, 11, 2
30	24, 19, 10, 25, 23	10, 24, 20, 22, 3
35	19, 25, 29, 34, 17	3, 13, 34, 26, 6
40	19, 30, 25, 38, 10	30, 10, 12, 24, 20

References

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