STATISTICAL ANALYSIS OF THE TECHNICAL CHARACTERISTICS OF ASYNCHRONOUS ROLLER-TABLE MOTORS

V. V. Dneprovsky, O. P. Muravlev, S. A. Shelekhov

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

A series of three-phase asynchronous motors with a squirrel-cage rotor, manufactured according to GOST 10283—62¹, is intended for operation in continuous and intermittent-duty modes. At present, motors of the AR series are considered among the most reliable electric motors used at metallurgical plants. They are capable of withstanding a short-circuit mode at rated voltage for about 10 min, whereas other asynchronous motors usually fail in this mode after 10–30 sec.

The quality of AR-series motors is characterized by observance of the accuracy of the parameters regulated by the standard. By accuracy in this case we shall mean the correspondence between the actual and the regulated parameters.

AR-series motors have been manufactured for 10 years. The rapid development of science and technology has made it necessary to update the AR series. To create a new series of roller-table motors satisfying world standards, scientifically substantiated technological and design development is required. Therefore, analysis of the actual scatter of motor parameters under the existing technology of their manufacture is necessary for the justified assignment of tolerances and nominal parameter values. In this connection, the present work gives the results of a statistical analysis of parameters obtained during acceptance and type tests of AR-series motors of frame sizes 4—7.

As a result of carrying out type and acceptance tests at SKBEM and at the “Sibelektromotor” plant, extensive statistical material has been accumulated, representing the AR series quite fully. This makes it possible to use probabilistic methods of analysis to investigate the scatter of parameters.

The quality of electric motors of the AR series is characterized by the accuracy of maintaining the following characteristics: starting torque \(M_{\text{p}}\), starting \(I_{\text{k}}\) and rated \(I_{\text{n}}\) currents, dynamic constant \(D\), short-circuit losses \(P_{\text{k}}\), rotational speed \(n_{\text{n}}\), efficiency \(\eta\), and power factor \(\cos \varphi\), which we shall henceforth call output parameters. The output parameters of electrical machines have scatter caused by a whole series of reasons

(technological deviations, quality of the starting materials, changes in temperature, measurement errors, etc.), and therefore they can be regarded as random variables having their own distribution law. For investigations it is necessary to know this distribution law of the parameters.

In works [1—3] it is shown that the output parameters of asynchronous motors and the parameters of control tests are random variables with a normal distribution law. The normality of the distribution curve can be judged from the values of the statistics characterizing skewness and kurtosis, which are calculated on the basis of experimental data. The criterion for normality of the distribution is the equalities [4]:

\[ a = l = 0, \]

where \(a\) is the empirical measure of skewness, and \(l\) is the empirical measure of kurtosis of the distribution.

If the empirical measures of skewness and kurtosis are such that the intervals \(a \pm 3\sigma_a\) and \(l \pm 3\sigma_l\) contain the zero point, then the distribution of the parameters may be considered normal. Here \(\sigma_a\) and \(\sigma_l\) are the standard errors of \(a\) and \(l\), determined by the formulas:

\[ \sigma_a = \sqrt{\frac{6}{n}}, \tag{2} \]

\[ \sigma_l = 2\sigma_a, \tag{3} \]

where \(n\) is the sample size.

The values of the statistics \(a\) and \(l\) and their standard errors, characterizing the distribution law, are calculated from the results of control tests of the motors.

Table 1 gives the values of the statistics for a number of parameters of motors AR 53-12 and AR 74-12. Analysis of Table 1 shows that, for the majo-

Table 1

rity of the parameters the normality condition (1) is satisfied, with the exception of \(P_0\) and \(M_p\) for AR 74-12, whose distribution may be considered approximately normal.

On the basis of Table 1 and previously conducted investigations on asynchronous motors of the A and AO, AO2 and VAO series, it may be considered that all the remaining output parameters of roller-table motors have a normal distribution law.

For a sufficiently reliable estimate of normally distributed random variables (the output parameters of electric motors), from experimental data we calculate the mean value of the output parameter \(\overline{X}\) and the corrected statistical variance \(\overline{S}^2\).

\[ \overline{S}^{2}=\frac{n}{n-1}\sigma^{2}, \tag{4} \]

where \(\overline{\sigma}\) is the root-mean-square deviation.

The corrected statistical variance \(\overline{S}^{2}\) is a consistent and unbiased estimate of the variance of the general population [4].

Tables 2 and 3 present the mean values of the output parameters and compare them with the specified (and unspecified) parameter values according to GOST 10283—62.

The same tables also give the new parameter values adopted in GOST 10283—69.

The data of Tables 2 and 3 show that the ratio between the actual mean values of the output parameters and the nominal values specified by GOST differs for different parameters and motor frame sizes. Thus, for example, with respect to \(I_k\), all motor frame sizes have a mean value below the specified value, and for a number of motors (AR53-8, 10; AR73-10, 12; AR74-10, 16) this discrepancy is significant.

The actual mean values of \(I_k\), obtained as a result of statistical processing, show that for a number of motor types the nominal values of \(I_k\) can be reduced. The large reserve built into production with respect to \(I_k\) is unjustified.

The output parameter \(P_k\) has a mean value, for all motor types, that approximately corresponds to the catalog value.

The actual values of \(M_p\) for all motor frame sizes are higher than the specified values, and for a number of motors (AR42-4; AR 43-4,6 and AR74-10,16) they exceed the catalog values by 30–50%. Moreover, in most cases a fairly clear pattern can be observed: starting torques in one and the same frame size increase as the number of poles increases. This is explained by the fact that the AR series was designed with maximum unification of parts and assemblies. The requirements stipulated in the technical specification limited the lower boundary of the values of \(M_p\) at low frequencies, which led to the actual values at 50 Hz being overestimated in comparison with the nominal values.

From the results of the comparison and analysis of the data in Table 2 for starting torques, it is clear that, with maximum unification of parts and assemblies, a fixed scale of starting torques for all rotational speeds at 50 Hz in one and the same frame size is unjustified.

Comparison of the mean values of the dynamic constant \(D\) at duty cycle \(PV=25\%\) with the values according to GOST 10283—62 and GOST 10283—69 was carried out for an excess stator-winding temperature of \(120^\circ\text{C}\). The recalculation was performed by the formula (GOST 10283—69)

\[ D_{120^\circ}=\frac{120}{\tau}D_{\tau}, \tag{5} \]

where \(D_{\tau}\) is the dynamic constant for a temperature \(\tau\) differing from 120 by \(\pm 5^\circ\text{C}\).

The values \(D_{120^\circ}\) given in GOST 10283—69 agree with the data obtained by statistical processing, although for motors AR52-12 and AR73-10 there is a possibility of increasing the nominal values of \(D_{120^\circ}\).

The mean values of \(I_n\), \(n_n\), \(\eta\), and \(\cos\varphi\) have deviations in both directions. Since the deviations are small, it may be considered that these parameters correspond to the established nominal values.

To assess the accuracy with which the parameters are ensured under the existing technology, values of the unbiased estimate of the variance \(\overline{S}^{2}\) were obtained.

Table 2

Table 3

As is known, for the normal distribution law, random variables with probability 0.9973 fall within the limits \(\pm 3\bar S\).

We shall estimate the scatter of parameters by means of half the scatter field

\[ \delta_x=\frac{300\bar S}{\bar x},\ \%. \tag{6} \]

Table 4 gives the values of half the scatter field for a number of parameters of asynchronous roller-table motors of the AR series, frame sizes 4–7.

Table 4

where

\(\Delta t_{c.u.}\) — temperature rise of the stator winding in continuous operating duty;

\(\Sigma P\) — total losses.

As can be seen from Table 4, some of the parameters have a large scatter, indicating that they are most sensitive to deviations of the input parameters.

Table 5 gives the average values of half the scatter field \(|\delta_x|\) and the tolerance according to GOST for a number of parameters of asynchronous roller-table motors of the AR series, frame sizes 4–7. For \(\eta\) and \(\cos\varphi\), the average tolerance was calculated.

Table 5

From Table 5 it is evident that the actual average value of the scatter of motor parameters is greater than that specified by GOST.

Analysis of the parameters from experimental data for motors of the AR series shows that, although for some indicators there is a considerable margin in comparison with the regulated values, at present large scatters limit the possibility of increasing the nominal values of output parameters to the average actual values. Taking account of the accuracy of the technology will make it possible to approach more soundly the limitation of scatters and the assignment of the nominal parameters themselves.

The scatter of output parameters is caused by fluctuations of the input parameters: the principal dimensions with their tolerances, the stability of the characteristics of the materials used, the manufacturing technology of the motors, the quality of assembly, and a large number of other random factors. Therefore the determination of input parameters must also be

statistical. In order to reduce the scatter of the output parameters, it is necessary to assess the proportional influence of each input parameter. The study of the input parameters and the determination of their proportional influence will make it possible to establish an economically justified minimum scatter of parameters that should be ensured by tightening control or by changing the manufacturing technology.

The analysis carried out of the output parameters, based on the results of type and control tests of AR-series motors by methods of mathematical statistics, makes it possible to draw the following conclusions:

1. The distribution of the output parameters obeys the normal law.

2. For some indices there is a considerable margin in comparison with the regulated values; however, at present, large scatters limit the possibility of increasing the nominal values of the output parameters to the actual mean values.

3. In order to reduce the field of dispersion of the output parameters, it is necessary to establish their interrelation with the input parameters and to assess the proportional influence of the scatter of each output parameter, so as to create a rational, economically justified system of tolerances, technology, and control. This will make it possible to bring the nominal values of the output parameters closer to the actual mean values and to make fuller use of the possibilities embodied in the design, thereby raising the technical level of the AR-series motors.

4. The rigid scale of initial starting torques adopted in the AR series is not rational. With maximum unification of the units and parts of motors of different pole numbers within one and the same frame size, it is more expedient to establish a non-rigid scale, which makes it possible to increase the initial starting torques of multipole motors.

REFERENCES

1. T. G. Soroker, O. D. Goldberg. Statistical quality control of asynchronous electric motors in series production. Vestnik elektropromyshlennosti, 1956, No. 5.

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

3. B. I. Burshtein, O. P. Muravlev, E. K. Strelbitskii. Scatter of the parameters of asynchronous motors with power from 1 to 70 W. Izvestiya TPI, vol. 190, 1968.

4. A. K. Mitropolskii. Technique of Statistical Computations. Fizmatgiz, Moscow, 1961.

5. V. E. Gmurman. Introduction to Probability Theory and Mathematical Statistics. Vysshaya shkola, Moscow, 1966.

6. S. A. Shelekhov. Study of asynchronous motors for driving roller tables. Candidate dissertation, Tomsk, 1967.

1. As of January 1, 1970, this GOST was replaced by the 1969 edition (GOST 10283-69). ↩