Preface

I originally planned to continue to derive the order and frequency of electromagnetic force according to the classification of radial force. When analyzing the spatial order of the magnetomotive force generated by the stator armature winding, the performance of integer slots and fractional slots was inconsistent. So I was trapped by the difference between integer slots and fractional slots in the stator, so I first learned the difference between the two in spatial order. Once this point is made clear, the subsequent theoretical derivation will be more confident. The content link of the previous article is as follow

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Integer slots and fractional slots are distinguished by the number of slots per pole and per phase:

Q is the number of slots in the motor, p is the number of pole pairs, and m is the number of phases. If q is a fraction, it is called a fractional slot motor, and if it is an integer, it is an integer slot motor.

Problem introduction

In the book “Analysis and Control of Motor Noise” written by Professor Zhu Ziqiang, it is believed that the number of stator harmonics generated by fractional slot windings is different from that generated by integer slot windings. The corresponding expressions are given respectively, but unfortunately there is no derivation process. In other literature on motor vibration noise, some do not distinguish the values ​​of stator harmonics, and use n for substitution derivation (consistent with my previous blog). Although the other part gives the value, it uses the expression of the magnetic motive force generated by the current of the ideal full-pitch winding, without distinguishing between integer slots and fractional slots. There are literatures that compare the two, but they do not derive mathematically, and directly give simulation and experimental results. I will continue to read more literature in this area. When writing this blog, I found some literature published by Professor Zhu Ziqiang and discussed this issue. I hope it will be inspiring.

I am very confused by the existing literature. Since everyone agrees that there is a difference between fraction slots and integer slots, why do some avoid discussing it while others only touch upon it briefly? Anyway, the issue is not explained clearly.

Thoughts

You can tell from the subtitle that I don’t understand it either (covering my face). But I have some understanding in this regard. The main purpose of this blog is to integrate my thoughts, otherwise my thoughts will be messed up again.

Let me explain my understanding through two examples, one is an 8-pole 9-slot motor and the other is an 8-pole 48-slot motor. Why is it said that the harmonics of the fractional slot motor are large, while the air gap magnetic field sinusoidality of the integer slot motor is better than that of the fractional slot motor? I previously focused on the stator and ignored that the overall magnetic field of the motor is actually generated by the interaction between the stator and the rotor, so I fell into a bottleneck.

First, the distribution of the motor magnetic field is given when only the stator is energized. At this time, the permanent magnet material is set to air.

 8 poles 9 slots 

 8 poles 48 slots

The difference in the absolute value of the magnetic flux density between the two is caused by the different number of turns given when the two motors were designed. Judging from the distribution of the magnetic flux density of the two, the fractional slot motor with a smaller number of slots has windings that are symmetrically distributed in space and are distributed in a certain area of ​​the motor . When a three-phase symmetrical current is passed through, the magnetic flux density in a certain spatial area of ​​the motor will be large, while the magnetic flux density in the remaining area will be small, that is, the overall magnetic flux density distribution of the motor is uneven. For the integer slot motor, although the windings also show a symmetrical distribution in space, they are evenly distributed in the entire area of ​​the motor , so the magnetic flux density distribution of the motor is relatively uniform.

The intuitive feeling is that the magnetic flux density of the fractional slot is unevenly distributed, while the integer slot looks more pleasing to the eye. The following are the waveforms of the magnetic flux density of the two with spatial position (mechanical angle).

8 poles 9 slots 

8 poles 48 slots

The waveform result is consistent with the magnetic flux distribution we see. The magnetic flux distribution of an integer slot is highly periodic, while that of a fractional slot is unsatisfactory. This result should not be viewed only from the stator. We say that the magnetic flux sinusoidality of the fractional slot motor is not good, not because the magnetic flux waveform generated on its stator side is not periodic, but because the magnetic motive force on its stator side is not uniform in space and does not correspond to the rotor magnetic field. When it is superimposed with the magnetic potential generated on the rotor side, the comprehensive waveform is not good-looking. This superimposed magnetic field is what we really care about when analyzing vibration and noise problems.

Next, let’s take a look at the waveform after the two are superimposed:

 8 poles 9 slots 

 8 poles 48 slots

These two pictures are the original data simulated by finite element method without any data processing. The vertical axis units are different, so don’t look at the absolute value of the magnetic flux density. The main purpose is to observe the waveform. The horizontal axis is still the spatial mechanical angle. The comparison results of these two pictures are very obvious. The same is an 8-pole motor. The spatial magnetic flux density distribution of 8 poles and 9 slots, due to the mismatch between the stator magnetic field and the rotor magnetic field distribution, the waveform periodicity is very poor after the two are superimposed. After the magnetic field of 8 poles and 48 slots is superimposed, the magnetic flux density distribution under each pole is almost exactly the same.

Finally, an FFT is performed on the original data for spatial angles, as shown below:

 8 poles 9 slots 

 8 poles 48 slots

The DC component is not discussed here. The difference in sinusoidal properties between the two can be seen from the FFT results of the two. The 8-pole 9-slot has some low-order harmonics due to the poor periodicity of the waveform. The main harmonics are concentrated on multiples of 8 and 9. The multiples of 9 are because the number of stator slots is 9 and the spatial distribution of the stator magnetic field and the rotor magnetic field is inconsistent, so there are harmonics of multiples of 9 after decomposition. In contrast, the FFT results of the 8-pole 48-slot are much clearer. The stator and rotor magnetic fields are distributed consistently, and the number of poles is 8, so the harmonics are all multiples of 8.

Final Words

This blog does not give a conclusive opinion, but only records the phenomenon observed so far. My current opinion is that since the ultimate concern is the superimposed magnetic field, it is not very meaningful to discuss the harmonic situation of the stator side magnetic field of the fractional slot motor alone. I will continue to discuss this point of view with teachers and students, and strive to express this part of the content clearly.