STATISTICAL STUDY OF THE PARAMETERS OF AO2 SERIES ASYNCHRONOUS ELECTRIC MOTORS

E. A. Knipenberg, S. N. Marukhin

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

Statistical methods for analyzing the parameters of asynchronous electric motors make it possible to refine catalog data, establish whether the scatter of parameters conforms to GOST tolerances, evaluate the accuracy of calculation, etc.

In the present work, some results are given of a statistical study of parameters obtained during factory tests of three-phase squirrel-cage asynchronous electric motors of the AO2-1÷4 series, in the basic design. The electric motors were manufactured by the Sibelektromotor plant and by the experimental shop of SKBE (Tomsk) in 1960–1968.

By parameters here are meant: the stator phase current \((I_0)\), losses \((P_0)\), and power factor \((\cos \varphi_0)\) at no load; the stator phase current \((I)\), efficiency \((\eta)\), power factor \((\cos \varphi)\), torque \((M)\), and slip \((S)\) in the rated operating mode; the starting \((M_k)\), minimum \((M_{\min})\), and maximum \((M_{\max})\) torques during starting; the initial starting current of the stator phase \((I_k)\); the short-circuit losses \((P_k)\) at rated voltage, etc. These quantities are determined during type tests and characterize the principal operating modes of the electric motor.

The actual values of the parameters of asynchronous electric motors exhibit scatter caused by technological deviations, measurement errors, and other random factors, and therefore may be regarded as random variables having their own distributions. It is important to establish the distribution law.

Analysis of the manufacturing and testing conditions of the electric motors under study, as well as data for other types of electric motors [1÷4], makes it possible to assume that the distribution law of the parameters is normal.

The criterion for normality of the distribution is the equalities [5]:

\[ \alpha = \iota = 0, \tag{1} \]

where

\[ \alpha \text{ — empirical measure of skewness,} \]

\[ \iota \text{ — empirical measure of kurtosis of the distribution.} \]

The distribution of the parameters may be considered normal if the deviation from zero of the quantity \(\alpha\) does not exceed \(3\sigma_{\alpha}\), and that of the quantity \(\iota\) does not exceed \(3\sigma_{\iota}\). Here \(\sigma_{\alpha}\) and \(\sigma_{\iota}\) are found from the formulas [5]:

\[ \sigma_{\alpha} = \sqrt{\frac{6}{n}}; \tag{2} \]

\[ \varepsilon_t = 2\sigma_{a}, \tag{3} \]

where \(n\) is the number of motors of one frame size investigated. Table 1 gives the values of the indicated quantities for a number of parameters of AO2-32-4 electric motors manufactured with the same winding data. \(\alpha\) and \(t\) were determined by the method of [5] on the “Promin-M” digital computer.

Table 1

As can be seen from Table 1, the deviation \(\alpha\) for the motor parameters under consideration does not exceed, in absolute value, \(3\sigma_\alpha\) (the interval \(\alpha \pm 3\sigma_\alpha\) contains zero), with the exception of \(\cos\varphi_0\), for which this deviation is \(3.22\sigma_\alpha\). The deviation \(t\) also does not exceed \((3\sigma_t)\), with the exception of \(P_k\), for which it is equal to \(3.5\sigma_t\).

Thus, the distribution of the overwhelming majority of the considered parameters of the AO2-32-4 asynchronous motor is normal, while the distribution of \(P_k\) and \(\cos\varphi_0\) is approximately normal. This conclusion agrees with data for other asynchronous motors: types A and AO [1, 3], KAMO-133-2 [2], and UAD [4]. Consequently, it may be stated that, for a fairly broad class of asynchronous electric motors, the distribution of the parameters is normal or approximately normal.

The normal distribution law of a random variable is completely determined by two characteristics—the mathematical expectation and the variance. Since these characteristics are unknown to us for the random variables (the parameters of electric motors), we shall use their estimates computed from experimental data. The estimates must be consistent, unbiased, and efficient [6]. For the normal law, the computed mean value \(\overline{X}\) satisfies these requirements. If the “corrected” statistical variance \(D\) [6] is taken as the measure of dispersion, then it will be a consistent, unbiased, and “asymptotically efficient” estimate.

We shall use the mean values to compare the actual parameters of the electric motors (according to the results of type tests) with the standard parameters (determined by the technical specifications).

Table 2 gives the results of the statistical processing of a number of parameters for 26 frame sizes of AO2-1÷4 gab. series electric motors. The values according to the technical specifications [7], in accordance with

Table 2

by which the motors under study were manufactured, and according to the catalog [8]. The multiplicities of the initial starting current and torque in Table 2 were obtained by dividing the mean values of the corresponding quantities.

Table 2 shows that, in terms of efficiency, most types of electric motors have actual values lower than the catalog values. In terms of $\cos \varphi$, the actual values deviate in both directions from the catalog values. Since the deviations are small, it may be considered that the parameters $\eta$ and $\cos \varphi$ of AO2-1–4-frame electric motors approximately correspond to the catalog values.

The ratio of the maximum starting torque to the rated torque is higher than the catalog value for all electric motors. The largest ratio is found in frame-size 4 electric motors, with the exception of AO2-41-4. For these electric motors, on average, it is 1.35 times higher than the catalog value, whereas for the other electric motors it is only 1.10 times higher.

The ratio of the minimum starting torque to the rated torque is significantly higher than the catalog value for all electric motors. Moreover, a fairly clear tendency is observed: as the number of poles increases, the deviation of the actual ratio of the minimum torque to the rated torque from the catalog value decreases. Thus, for 2-pole electric motors this ratio is greater than the catalog value, on average, by a factor of 2.26; for 4-pole motors, by a factor of 1.58; for 6-pole motors, by a factor of 1.40; and for 8-pole motors, by a factor of 1.32.

The ratio of the initial starting torque to the rated torque for electric motors AO2-31-6, AO2-32-6, and AO2-41-4 is less than the catalog value; for the remaining electric motors it is greater than the catalog value. The largest deviation is found in 2-pole electric motors of all frame sizes. For them, the actual ratio of the initial starting torque to the rated torque is, on average, 1.29 times greater than the catalog value.

Finally, the ratio of the initial starting current to the rated current for electric motors AO2-32-2 and AO2-42-2 is greater than the catalog value, while for the remaining electric motors it is less than the catalog value. Moreover, the deviation of the actual ratio of the starting current to the rated current from the catalog value decreases as the frame size increases.

Thus, even such a preliminary analysis shows that the actual parameter values of asynchronous electric motors of the AO2 series, frame sizes 1–4 (mean values obtained by statistical processing), differ from the catalog values, and for some parameters quite substantially.

In the statistical processing of the results of type tests of the electric motors under study, in addition to the mean parameter values, we obtained values of the unbiased estimate of the variance $\overline{D}$. For the normal distribution law, as is known, all parameter values obtained in practice fall, with probability 0.9973, within the limits

\[ \pm 3\sqrt{\overline{D}}. \]

Consequently, these limits can be used with sufficient accuracy to estimate the scatter of the parameters about the mean value $\overline{X}$. For convenience, we shall estimate the scatter by half of the dispersion field:

\[ \delta_x = \frac{300\sqrt{\overline{D}}}{\overline{X}},\ \%. \tag{4} \]

Table 3 gives the limiting values of $\delta_x$ according to (4) for a number of parameters of asynchronous electric motors of the AO2 series, frame sizes 1–4, comprising 26 type sizes.

Table 3

The greatest scatter is exhibited by the minimum starting torque $M_{\min}$. The scatters of the slip in rated operation $S$, the no-load losses $P_0$, and the no-load power factor $\cos\varphi_0$ are also large.

From $\delta_x$ one can assess the use of the tolerances established by GOST 183—66 for the parameters of asynchronous electric motors. Table 4 gives the mean values of half the scatter field $\overline{|\delta_x|}$ and the tolerance according to GOST for a number of parameters of asynchronous electric motors of the AO2-1–4 frame-size series. The tolerance for $\eta$ and $\cos\varphi$ is given as an average.

Table 4

It is evident from Table 4 that, for efficiency, power factor, and maximum torque, the actual scatter (on average) is less, and for the initial starting current it is significantly less, than the tolerance according to GOST. For the initial starting torque, the actual scatter approximately corresponds to the tolerance. Despite the large tolerance for the minimum starting torque and the slip in rated operation (respectively −20% and +25% of the rated value), the actual scatter of these parameters is above the tolerance (by 30% on average).

It should be noted that, for a final judgment on whether the actual scatter of the parameters conforms to the tolerances according to GOST, it is necessary to carry out analogous studies for electric motors manufactured at other plants in the USSR.

Conclusions

For the electric motors of the AO2-1–4 frame-size series studied, on the basis of an analysis of the results of type tests by statistical methods, the following conclusions may be drawn:

1. The distribution of the parameters of the electric motors is normal or approximately normal.

2. The actual mean values of the parameters differ from the catalog values, and for some parameters—significantly.

3. The use of the tolerances according to GOST is very nonuniform for different parameters.

The results of the statistical processing of the parameters, presented in Tables 1–4, may be used as refined reference material in various calculations and studies.

REFERENCES

1. T. G. Soroker, O. D. Gol'dberg. Statistical quality control of asynchronous electric motors in serial production. Vestnik elektropromyshlennosti, 1956, No. 5.

2. I. P. Isaev. Statistical evaluation of the stability of the characteristics of asynchronous motors. Elektrichestvo, 1957, No. 4.

3. O. P. Muravlev. Study of the influence of accuracy characteristics on the quality and reliability of asynchronous electric motors. Candidate’s dissertation, Tomsk, 1966.

4. B. I. Burshtein, O. P. Muravlev, E. K. Strel’bitskii. Scatter of the parameters of asynchronous motors with power from 1 to 70 W. Proceedings of TPI, vol. 190, 1968.

5. A. K. Mitropol’skii. Technique of Statistical Computations. Fizmatgiz, Moscow, 1961.

6. E. S. Venttsel’. Probability Theory. 3rd ed. Nauka, Moscow, 1964.

7. MRTU 2-015-002-62. Three-phase squirrel-cage asynchronous motors of the A2 and AO2 series with power from 0.6 to 100 kW.

8. Information 01. 01. 02—65. Three-phase alternating-current asynchronous electric motors with power from 0.6 to 100 kW. New unified A2 and AO2 series.