On the Influence of the Ratio Between the Numbers of Stator and Rotor Slots on the Uniformity of Rotation of Micromotor Rotors
Abstract Generated abstract
This paper examines how the ratio of stator to rotor slot numbers affects the uniformity of rotation in asynchronous micromotor rotors, a question previously studied mainly in relation to noise, vibration, and mechanical characteristic dips. Experiments on 30 W motors with 24 stator slots varied stator winding parameters and rotor slot numbers, measured instantaneous speed instability by a frequency inertial method, and analyzed the harmonic content of rotor speed oscillations. The results indicate that slot ratio and stator field harmonic composition substantially affect rotational nonuniformity, with only a few oscillation harmonics dominating the effect. For the tested 24 slot stator, rotor slot numbers 14, 17, and 31 were found more favorable for uniform rotation, while 11 was least favorable.
Full Text
ON THE INFLUENCE OF THE RATIO BETWEEN THE NUMBERS OF STATOR AND ROTOR SLOTS ON THE UNIFORMITY OF ROTATION OF MICROMOTOR ROTORS
V. B. Annenkov, A. S. Kurakin, N. V. Rybin
(Presented by the scientific seminar of the departments of electrical machines and general electrical engineering)
One of the main problems that arise in the creation of asynchronous micromotors is the choice of the ratio between the numbers of slots of the stator \(Z_1\) and the rotor \(Z_2\). The harmonic composition of the magnetic field in the air gap, which determines the overall level of alternating forces acting in the electric motor, depends to a large extent on this factor. These forces make starting the machines more difficult and also cause their noise and vibrations. A large number of works have been devoted to the choice of the ratio between \(Z_1\) and \(Z_2\) from the standpoint of low noise, absence of vibrations, and absence of dips in the mechanical characteristic. However, the development of new directions in technology compels us to return once again to consideration of the ratio between the numbers of stator and rotor slots in light of the requirements of high uniformity of rotor rotation.
As studies show, the curve of variation of the instantaneous rotational speed of micromachine rotors has the character of forced oscillations; therefore, the search for the causes of the occurrence of exciting forces is one of the principal questions in studying the physics of the process of rotor oscillations. In this connection, it is natural to search for the causes of such oscillations in the overall level of alternating forces acting in the electrical machine. This explains the presently widespread opinion that the noises, vibrations, and oscillations of the instantaneous rotational speed of the rotors of electrical machines have one and the same nature. Whereas the theory of the influence of the ratio between the numbers of slots \(Z_1\) and \(Z_2\) on noise and vibrations has been confirmed by numerous experimental proofs, no works have been published on the influence of this ratio on the uniformity of rotation of the rotors of asynchronous machines. This circumstance makes the conduct of such studies one of the urgent tasks.
In carrying out the experimental studies, the uniformity of rotor rotation was evaluated by the magnitude of the instability of their instantaneous speed, equal to [1]
\[ N \cdot \frac{n_{\max} - n_{\min}}{2n_0}, \]
where \(n_{\max}\), \(n_{\min}\), \(n_0\) are, respectively, the maximum, minimum, and mean values of the rotational speed of the rotor during its oscillations.
The quantity \(N\) was measured by the frequency-inertial method, which makes it possible to transform oscillations of the instantaneous rotational speed of shafts.
into an electrical signal. The form and magnitude of this signal completely reproduce the character of the rotor oscillations.
For the experimental investigations, asynchronous motors with a power of 30 W were used, having a stator with \(Z_1=24\). A three-phase winding was placed in each of the stators, the phases of which were connected in a star. The stator windings differed in the number of pole pairs and were structurally made in such a way as to allow the number of slots \(q\) per pole and phase to be changed without disassembling the motor. One of the rotors with \(Z_2=0, 11, 14, 17, 31\) could be inserted into each of the stators. The rotors had closed slots, which makes it possible to exclude the influence on the uniformity of their rotation of higher harmonics of the fields caused by rotor slotting. Thus, the selected method of conducting the investigations makes it possible to vary both the winding fields of the stator and the fields of the rotor reaction to the spectrum of the stator fields. The relative weight of the higher harmonics of the stator magnetomotive force was estimated by the harmonic coefficient, equal to
\[ k_{\mathrm{g}} = \frac{\sqrt{\sum_{\nu} F_{\nu}^{2} - F_{1}^{2}}}{F_{1}} \cdot 100\%, \]
where
\[ \begin{aligned} F_1 &\text{ is the effective value of the first harmonic,}\\ \sum F_\nu &\text{ is the effective value of the sum of the }\nu\text{-th harmonics of the stator winding m.m.f.,}\\ \nu &\text{ is the ordinal number of the higher harmonics of the m.m.f.} \end{aligned} \]
The value \(k_{\mathrm{g}}\) was calculated taking into account higher harmonics up to and including the 59th, since the amplitude of the remaining harmonics is very small in comparison with the fundamental wave of the m.m.f.
Leaving aside the widely known methods for determining the ordinal number and magnitude of the individual stator and rotor harmonics, we shall proceed directly to the experimental results. As the investigations show, the shape of the instability curve of the instantaneous rotational speed of asynchronous micromotor rotors differs from a sinusoid. Its harmonic composition depends on the ratio between the numbers of stator and rotor slots, and also on the relative weight of the higher harmonic m.m.f. of the stator winding. In carrying out the investigations, the rotor-oscillation curve was decomposed into a trigonometric series by means of a harmonic analyzer. Tables 1–6 present the results of such an analysis. The frequency of the higher harmonic components of the speed-oscillation curve is denoted by \(f_k\), its amplitude by \(N_k\), and the amplitude of the resultant curve by \(N_{\mathrm{kr}}\).
Table 1
| \(p=2\), \(q=1\), \(y=\tau\), \(k_{\mathrm{g}}=28,5\%\); \(f_k\) [Hz]; \(N_k \cdot 10^{-4}\) | \(z_2=0\) | \(z_2=11\) | \(z_2=14\) | \(z_2=17\) | \(z_2=31\) |
|---|---|---|---|---|---|
| 25 | 2,2 | 1,2 | 0,14 | 9 | 9 |
| 50 | — | 0,5 | — | 3,2 | — |
| 75 | — | 0,7 | 0,8 | 7 | 1,4 |
| 90 | 0,7 | 1,4 | 0,8 | — | 1,6 |
| 100 | 1,8 | 32 | 3 | 2,4 | 14 |
| 200 | — | 0,5 | 1,4 | 0,32 | 1,4 |
| \(N_{\mathrm{kr}}\cdot 10^{-4}\) | 5,8 | 39,2 | 5,9 | 23,5 | 29,2 |
Table 2
\(p=2\), \(q=2\), \(y=\tau\), \(K_{\mathrm{g}}=17,34\%\)
| \(f_k\) [Hz] / \(N_k \cdot 10^{-4}\) | \(Z_2=0\) | \(Z_2=11\) | \(Z_2=14\) | \(Z_2=17\) | \(Z_2=31\) |
|---|---|---|---|---|---|
| 25 | 2,8 | 1,1 | 2 | 2,2 | 7 |
| 50 | 1 | 0,3 | 1,6 | 4,2 | — |
| 90 | 0,4 | 1,4 | 0,2 | — | 0,4 |
| 100 | 0,39 | 28 | 1,6 | 0,8 | 3 |
| \(N_{\mathrm{kp}}\cdot 10^{-4}\) | 5,1 | 31,8 | 5,4 | 7,2 | 10,4 |
Table 3
\(p=2\), \(q=2\), \(y=\frac{5}{6}\tau\)
| \(f_k\) [Hz] / \(N_k \cdot 10^{-4}\) | \(Z_2=0\) | \(Z_2=11\) | \(Z_2=14\) | \(Z_2=17\) | \(Z_2=31\) |
|---|---|---|---|---|---|
| 25 | 2 | 16 | 1,8 | 7 | 11 |
| 36 | — | 0,6 | 0,8 | 0,2 | — |
| 50 | — | — | 0,11 | 0,4 | — |
| 75 | — | 0,9 | 0,5 | 0,5 | — |
| 100 | 2 | 0,3 | 1,2 | 0,1 | 3 |
| \(N_k\cdot 10^{-4}\) | 4,8 | 18,7 | 5,1 | 8,3 | 14 |
Table 4
\(p=2\), \(q=2\), \(y=\frac{2}{3}\tau\)
| \(f_k\) [Hz] / \(N_k \cdot 10^{-4}\) | \(Z_2=0\) | \(Z_2=11\) | \(Z_2=14\) | \(Z_2=17\) | \(Z_2=31\) |
|---|---|---|---|---|---|
| 25 | 1,6 | 15 | 2,8 | 8 | 11 |
| 32 | 0,6 | 1,8 | — | — | — |
| 50 | — | — | 0,18 | 0,6 | — |
| 100 | 0,12 | 0,24 | 1,14 | 1,6 | 1,7 |
| \(N_{\mathrm{kp}}\cdot 10^{-4}\) | 2,7 | 17,5 | 4,8 | 9,2 | 13,7 |
Table 5
\(p=1\), \(q=2\), \(y=\tau\), \(K_{\mathrm{g}}=13,3\%\)
| \(f_k\) [Hz] / \(N_k \cdot 10^{-4}\) | \(Z_2=0\) | \(Z_2=11\) | \(Z_2=14\) | \(Z_2=17\) | \(Z_2=31\) |
|---|---|---|---|---|---|
| 50 | 1 | 0,6 | 0,8 | 0,1 | 1,8 |
| 100 | 1 | 1,8 | 0,6 | 0,7 | 1 |
| 150 | 0,04 | 0,4 | 0,25 | 0,3 | 0,1 |
| 200 | — | 0,5 | 0,2 | 0,2 | 0,2 |
| \(N_{\mathrm{kp}}\cdot 10^{-4}\) | 2,1 | 3,7 | 2,1 | 2 | 3,4 |
Table 6
| \(p=1\) | \(q=4\) | \(y=3/4\tau\) | \(k_r=6\%\) | |
|---|---|---|---|---|
| \(f_k\,[\mathrm{Hz}]\) / \(N_k\cdot10^{-4}\) | \(Z_2=0\) | \(Z_2=11\) | \(Z_2=14\) | \(Z_2=17\) |
| 50 | 0,9 | 0,25 | 0,8 | |
| 100 | 1 | 1,2 | 0,45 | 0,75 |
| 150 | 0,8 | 1,8 | — | — |
| 200 | — | 0,1 | 0,2 | 0,18 |
| \(N_k\cdot10^{-4}\) | 3,1 | 3,8 | 0,9 | 2,15 |
Conclusions
-
The ratio between the numbers of stator and rotor slots in asynchronous electric motors has a substantial influence on the uniformity of rotor rotation. Through a favorable choice of this ratio and of the winding data of the stator winding, it is possible to improve the uniformity of rotor rotation by more than a factor of 10.
-
The degree of uniformity of rotation of the rotors of asynchronous micromotors is mainly affected by a small number of oscillation harmonics.
-
Changing the harmonic composition of the stator field at \(Z_2=0\) does not have a substantial influence on the uniformity of rotor rotation, despite the improvement in the dynamic properties of the motor due to a twofold increase in speed.
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For the given ratio between \(Z_1\) and \(Z_2\), improving the harmonic composition of the m.m.f. of the stator winding reduces both the overall level of nonuniformity of rotor rotation and the magnitude of individual oscillation harmonics. The number of higher harmonics in the oscillation curve also decreases.
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From the standpoint of obtaining high indices of uniformity of rotor rotation and optimal characteristics of the electric motor, the following numbers of rotor slots are most favorable for \(Z_1=24\): 14, 17, 31.
-
The number of rotor slots \(Z_2=11\) is the least favorable from the standpoint of rotational uniformity at \(Z_1=24\).
References
- A. S. Kurakin, V. B. Annenkov. Uniformity of rotation of synchronous micromotors. Elektrotekhnika, 1967, No. 2.